On the Hyperbolic Uniformization of Shimura Curves with an Atkin-Lehner Quotient of Genus 0 Joan Nualart Riera Aquesta tesi doctoral està subjecta a la llicència Reconeixement- NoComercial – CompartirIgual 4.0. Espanya de Creative Commons. Esta tesis doctoral está sujeta a la licencia Reconocimiento - NoComercial – CompartirIgual 4.0. España de Creative Commons. This doctoral thesis is licensed under the Creative Commons Attribution-NonCommercial- ShareAlike 4.0. Spain License. Joan Nual artR iera Onth eHy perbo licU niform izatio nofS himu raCu rves with anA tkin- Lehn erQu otien tofG enus 0 Joan Nualart Riera On the Hyperbolic Uniformization of Shimura Curves with an Atkin-Lehner Quotient of Genus 0 Departament d’Àlgebra i GeometriaUniversitat de Barcelona ON THE HYPERBOLIC UNIFORMIZATION OF SHIMURA CURVES WITH AN ATKIN-LEHNER QUOTIENT OF GENUS 0 Joan Nualart Riera Facultat de Matemàtiques Departament d’Àlgebra i Geometria Octubre de 2015 2010 Mathematics Subject Classification: 11F03, 11F12, 11G18, 14G35, 30F10 Descriptors o paraules clau: Corbes de Shimura, Uniformització hiperbòlica, Quocients d’Atkin-Lehner, Formes automorfes, Funcions automorfes, Invariants d’Igusa. Amb el finançament parcial d’una beca FPU del Ministerio de Educación y Ciencia i dels projectes MTM2006-04895, MTM2009-07024, MTM2012-33830. UNIVERSITAT DE BARCELONA Facultat de Matemàtiques Departament d’Àlgebra i Geometria ON THE HYPERBOLIC UNIFORMIZATION OF SHIMURA CURVES WITH AN ATKIN-LEHNER QUOTIENT OF GENUS 0 Memòria presentada per Joan Nualart Riera per a aspirar al grau de Doctor en Matemàtiques Departament d’Àlgebra i Geometria Facultat de Matemàtiques Universitat de Barcelona Barcelona, octubre de 2015 Universitat de Barcelona Programa de doctorat en Matemàtiques Doctorand: Joan Nualart Riera Director de tesi: Artur Travesa Grau Artur Travesa Grau, professor titular del Departament d’Àlgebra i Geometria de la Universitat de Barcelona, FA CONSTAR que aquesta memòria ha estat realitzada sota la seva direcció per Joan Nualart Riera i que constitueix la seva tesi per a aspirar al grau de Doctor en Matemàtiques. Barcelona, octubre de 2015 Signat: Artur Travesa Grau Als meus pares Contents Introduction v 1 Quaternion algebras, Shimura curves and automorphic forms 1 1.1 Quaternion algebras . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 Quaternion orders . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Shimura curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2.1 The upper half-plane . . . . . . . . . . . . . . . . . . . . . 11 1.2.2 Homographic transformations . . . . . . . . . . . . . . . . 12 1.2.3 Fundamental domains . . . . . . . . . . . . . . . . . . . . 13 1.2.4 Groups of quaternion units and the corresponding Rie- mann surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2.5 Complex Multiplication (CM) points . . . . . . . . . . . . 16 1.2.6 Ring Class Fields . . . . . . . . . . . . . . . . . . . . . . . 17 1.2.7 The Shimura curve X(D,N) . . . . . . . . . . . . . . . . 20 1.3 Automorphic forms . . . . . . . . . . . . . . . . . . . . . . . . . . 23 2 Kroneckerian polynomials 27 2.1 The polynomials ΨΓ,N,f . . . . . . . . . . . . . . . . . . . . . . . 28 2.1.1 Relation to class equations . . . . . . . . . . . . . . . . . 39 2.1.2 Singularities of the kroneckerian polynomials of prime power level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.2 Kroneckerian polynomials for the Hauptmoduln tD of X(D, 1) . 51 i ii 3 The curves X(D, 1) of genus 0 65 3.1 Fundamental domains for Γ(D,N) . . . . . . . . . . . . . . . . . 66 3.2 Some explicit fundamental domains in the cases D = 6, 10, 22 . . 71 3.3 q-expansions of automorphic functions . . . . . . . . . . . . . . . 79 3.3.1 Solutions of automorphic differential equations by means of order 2 linear differential equations . . . . . . . . . . . 84 3.4 Hauptmoduln: t+D . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 3.4.1 An overview of the case D = 6 . . . . . . . . . . . . . . . 85 3.4.2 The cases D = 10 and D = 22 . . . . . . . . . . . . . . . . 87 3.5 Expression of t+D in terms of absolute Igusa invariants . . . . . . 93 3.5.1 From a point on X(D, 1) to the corresponding abelian surface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 3.5.2 The case of t+6 . . . . . . . . . . . . . . . . . . . . . . . . 99 3.5.3 The case of t+10 . . . . . . . . . . . . . . . . . . . . . . . . 100 3.5.4 The case of t+22 . . . . . . . . . . . . . . . . . . . . . . . . 101 4 Uniformization of some Shimura curves with full Atkin-Lehner quotient of genus 0 115 4.1 All the curves X(D,N) with an Atkin-Lehner quotient of genus 0 116 4.2 A detailed case: the curve X(22, 3) . . . . . . . . . . . . . . . . . 135 4.3 Other cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 4.3.1 The curve X(22, 5) . . . . . . . . . . . . . . . . . . . . . . 147 4.3.2 The curve X(22, 7) . . . . . . . . . . . . . . . . . . . . . . 151 4.3.3 The curve X(6, 5) . . . . . . . . . . . . . . . . . . . . . . 155 4.3.4 The curve X(6, 7) . . . . . . . . . . . . . . . . . . . . . . 161 4.3.5 The curve X(6, 11) . . . . . . . . . . . . . . . . . . . . . . 164 4.3.6 The curve X(10, 3) . . . . . . . . . . . . . . . . . . . . . . 168 4.3.7 The curve X(10, 7) . . . . . . . . . . . . . . . . . . . . . . 171 4.3.8 The curve X(10, 9) . . . . . . . . . . . . . . . . . . . . . . 174 5 Automorphic form expansions: arithmetic properties 181 5.1 Shimura-Maaß operators and coefficients . . . . . . . . . . . . . . 182 iii 5.2 Arithmeticity principle . . . . . . . . . . . . . . . . . . . . . . . . 185 5.2.1 Shimura reciprocity . . . . . . . . . . . . . . . . . . . . . 192 6 Computational applications: kroneckerian polynomials and ex- pansions of automorphic functions and forms 195 6.1 Computation of kroneckerian polynomials . . . . . . . . . . . . . 196 6.1.1 Examples in the case D = 6 . . . . . . . . . . . . . . . . . 199 6.2 Computation of the values of a Hauptmodul and its expansions at a given CM point . . . . . . . . . . . . . . . . . . . . . . . . . 206 6.2.1 All rational and quadratic CM points on the curvesX(D, 1)+ for D = 6, 10, 22 . . . . . . . . . . . . . . . . . . . . . . . 208 6.3 Computation of the local parameters . . . . . . . . . . . . . . . . 222 6.4 Automorphic forms, Hecke operators and q-expansions . . . . . . 231 6.4.1 Cusp forms attached to Shimura curves of discriminant 6 240 6.4.2 Cusp forms attached to Shimura curves of discriminant 10 249 6.4.3 Cusp forms attached to Shimura curves of discriminant 22 255 Resum en català 269 Bibliography 287 Introduction The main goal of this thesis is to contribute to the explicit hyperbolic uni- formization of Shimura curves. We will restrict to the case of curves attached to Eichler orders in rational quaternion algebras whose maximal Atkin-Lehner quotient has genus 0, which despite multiple differences bears some resemblance to the classical modular case. We will provide an approach to obtain an explicit uniformization of these curves and some of their covers, together with several applications. We will illustrate all the applications with plenty of examples. In order to get a better understanding of the problem, we will begin by introducing some historical background. Then, we will explain in detail our contributions and the content of the monograph. The fuchsian groups and the corresponding automorphic functions are an important topic of study in the mathematics of the 19th century, which reaches a culminating point with the works of Poincaré [Poi95], Klein [Kle23, Kle79], Fricke [Fri92, Fri93, Fri94, Fri99] and Fricke and Klein [KF90, KF92, FK97, FK12]. However, some particular cases of fuchsian groups and automorphic functions, for example those attached to congruence subgroups Γ0(N) of the modular group Γ0(1) = SL(2,Z), can be traced back to the works of Gauss, Abel or Dedekind. It is in this setting that the most well-known automorphic function makes its appearance, namely the j function (or j invariant), commonly called Klein j function, but which was already introduced, up to scaling, by Dedekind [Ded77] as the function Valenz, and that, according to Klein [Kle79], stems from the works of Gauss. The j function is an automorphic function for the modular group SL(2,Z), and therefore, as for all subgroups Γ0(N), is traditionally called a modular elliptic function. If Γ is a fuchsian group such that Γ\H has finite hyperbolic volume (or more generally a fuchsian group of the first kind) and we consider its action v vi Introduction on the upper half-plane, then the automorphic functions with respect to Γ are the meromorphic functions on H which are invariant under the action of Γ and which satisfy certain growth conditions. On the other hand, the action of Γ on H induces a structure of Riemann surface on the quotient space Γ\H and it allows to identify Γ\H with a Zariski open subset of the complex points of an algebraic curve. In particular, for the case of congruence subgroups of the modular group, this gives rise to the classically called modular curves X0(N). The field of rational functions of the modular curve X0(1) can be nat- urally identified with C(j). An impor- tant characteristic of the Riemann sur- face Γ0(N)\H is that it is not compact. For example, if N = 1, a fundamental domain for the action of SL(2,Z) on H is the well-known Gauss’ fundamental domain, which originally appeared in the context of the reduction of binary quadratic forms. In particular, this al- lows to consider Fourier expansions for the modular functions. i ρ i∞ b b a a For example, for the case of the j function, the Fourier expansion yields the well-known expression j(τ) = 1 q + 744 + 196884q + 21493760q2 + . . . where q = e2πiτ . In particular, the rationality (and integrality) conditions of the coefficients of these expansions translate into properties of the curves X0(N). For example, the fact that Γ0(N) = ￿ N 0 0 1 ￿−1 SL(2,Z) ￿ N 0 0 1 ￿ ∩ SL(2,Z) allows us to write C(X0(N)) = C(j, j(N ·)). Moreover, there exists a monic sym- metric polynomial of degree ψ(N) in each variable, ΦN (X,Y ) ∈ Z[X,Y ], such that ΦN (j, j(N ·)) = 0, which provides an equation for X0(N) and is classically called the modular equation of level N . This polynomial has roots (j, j ◦ αN ), vii where αN runs over the transformations corresponding to a set of representa- tives of primitive elements of norm N inM(2,Z) modulo the action of SL(2,Z). This allows to explicitly compute ΦN (X,Y ) using the Fourier expansion of j, since it is possible to choose a complete set of representatives αN fixing ∞. In addition, the polynomial ΦN (X,X) can be factored into smaller pieces known as class equations, each of those corresponding to the values of the j function at certain points. In this way, the values of the j function at a set of points in the upper half-plane known as complex multiplication (CM) points, which consists of the quadratic points in H, can be determined. The value of j at such a point τ is not only an algebraic number but it generates a certain abelian ex- tension of the field Q(τ). In particular, this provides a way to explicitly obtain polynomials defining these extensions. On the other hand, it is well-known that the points of X0(1) correspond to isomorphism classes of generalized elliptic curves. A point τ ∈ SL(2,Z)\H corresponds to the isomorphism class of the elliptic curve represented by C/￿1, τ￿ and the coefficients of a Weierstrass model for it are given by means of modular elliptic forms. Then, the function j is a rational function of the coefficients of the Weierstrass model. Hence, from an algebraic point of view, the j function can be identified with an absolute invariant of binary cubic (or quartic) forms which classifies isomorphism classes of elliptic curves. Many more properties of the modular functions have been studied; for exam- ple: the reduction of the polynomial Φp(X,Y ) modulo p, which is given by the Kronecker congruence formula; some alternative functions generating the same abelian extensions generated by j at the CM points, which give to rise simpler defining equations, like Weber modular functions, etc. Modular forms, as we have seen, also appear in this context, either in the form of theta functions (Ja- cobi) or intervening in the expansions of modular functions (Eisenstein), even though they were only later formalized by Hecke [Hec24]. It was also Hecke who introduced the L-series attached to an eigenform and proved some of its impor- tant properties, like the existence of an Euler product expansion. Introducing in this way a notion which has played an important role in some of the most important results in number theory since the second half of the 20th century. However, not only automorphic functions and forms for congruence sub- groups of SL(2,Z) have been systematically considered. For example, one of the first generalizations which was introduced is that one corresponding to the Hilbert modular group SL(2,OK), where Z is replaced by the ring of integers of a totally real number field, giving rise in this way to Hilbert modular functions and forms. viii Introduction Similarly, if we replace the upper half-plane by the set of all square matrices of dimension g over C with positive definite imaginary part and consider the group Sp(2g,Z) acting on it, we obtain Siegel modular functions and forms of genus g, introduced by Siegel [Sie35, Sie39] after the work of Riemann [Rie57] on Riemann surfaces of genus g. For g = 1, this notion coincides with the classical modular case. For g = 2, a detailed study of the field of modular functions and the graded ring of modular forms was carried out by Igusa [Igu60, Igu62, Igu64, Igu67], including their interpretation as invariants of sextic forms. Generators in this case are known as Igusa invariants. It is in this context, and after the work of Eichler [Eic37, Eic38, Eic55a, Eic55b] on quaternion algebras, that Shimura [Shi67] considers the fuchsian groups obtained through an embedding from the elements of norm 1 in certain orders of a rational quaternion algebra H such that H ⊗Q R ∼=M(2,R), giving rise to the Shimura curves which we are going to study in this thesis. Shimura considers as well the case of quaternion algebras H over totally real number fields of degree d satisfying that H⊗R ∼=M(2,R)r×Hd−rR for some integer 0 < r ≤ d, giving rise to Shimura varieties of dimension r. And even more generally, in [Shi75a] the construction is extended to the case of reductive groups acting on bounded symmetric domains. This construction was later reinterpreted by Deligne [Del71, Del79], whose approach is commonly used nowadays. We will restrict to the rational case and we will assume that H ￿∼=M(2,Q). In this case we will consider an indefinite rational quaternion algebra of dis- criminant D, an embedding Φ : H ￿→M(2,R) and the group Γ ⊂ SL(2,R) (or its image in PSL(2,R)) corresponding to the image inM(2,R) of the elements of norm 1 of a suitable order O ⊂ H. An important difference between this case and the modular case is that in this situation the corresponding Riemann surface is always compact. In particular, there is no way to obtain Fourier expansions for the corresponding automorphic functions. Thanks to Shimura [Shi67], we know that there exists a canonical model for the Riemann surface attached to Γ, which is a projective curve X(Γ) defined over Q together with an analytic map jΓ : Γ\H → X(Γ)(C), giving an isomorphism with its image, with suitable conditions for its values at certain points (CM points). In order to do so, Shimura proves that X(Γ) is a coarse moduli space. More precisely, the points in X(Γ)(C) correspond to isomorphism classes of principally polarized abelian surfaces with quaternionic multiplication by a maximal order contain- ing O (which are usually called fake elliptic curves) together with a certain level structure. It is worth observing that the construction of Shimura is a general- ization which comprises the classical modular case and the function jΓ plays the ix role of the Klein j function. But, if H ￿∼=M(2,Q), it is proved by Shimura him- self [Shi75a] that any corresponding Shimura curve does not have real points. Therefore, the function jΓ will not just be an automorphic function in a suitable affine chart, but a tuple of several such functions. The arithmetical and diophantine properties of Shimura curves have been widely studied since its introduction in the 60s, for example: a p-adic uniformiza- tion was given by Čerednik [Čer76] and Drinfeld [Dri76]; the study of integral models, by Morita [Mor81], Boutot and Carayol [BC91], etc.; the study of local points, by Jordan and Livné [JL85], Ogg [Ogg85], etc.; the existence of rational points on certain (Atkin-Lehner) quotients, by Jordan [Jor81], Ogg [Ogg83], etc. In addition they have played an important role, together with modular curves, in the proof of some of the most important conjectures in number theory in the last times, essentially through the proof of the modularity of all elliptic curves over Q, thanks to the work of Eichler, Shimura and Wiles, among others. In this way, both modular and Shimura curves are coarse moduli spaces of (fake) elliptic curves and, at the same time, provide modular parametrizations for all the elliptic curves over Q. Note however that the function jΓ is not explicit in Shimura’s construc- tion. Even forgetting about the function jΓ, computing equations for the curve X(Γ) does not have an obvious approach and is in general a difficult prob- lem. The equation of a (non-modular) Shimura curve was first computed by Ihara [Iha79] and since then several other equations have been computed using different techniques, starting by Kurihara [Kur79], Jordan [Jor81] and Roberts [Rob89] and continuing with Elkies [Elk98], Gonzàlez and Rotger [GR04, GR06], Molina [Mol12], etc. Most of these authors make use of the Čerednik-Drinfeld uniformization to compute the equations. However, Elkies uses the complex uniformization provided by the Shimura canonical model. In this way, he re- covers as well the CM points corresponding to certain orders. A mixture of both approaches has been considered as well by Yang [Yan13] and Tu [Tu14], extending the cases computed by Elkies to the curves such that suitable covers of genus 0 exist, adding the knowledge of some equations for them, essentially those computed by Gonzàlez and Rotger mentioned before. In order to compute with automorphic forms and functions corresponding to Shimura curves attached to non-split quaternion algebras, some other type of power expansions, which replace the Fourier expansions usually used in the modular case, have to be considered. The fundamental ideas of the expansions considered can be traced back to Shimura [Shi75a], and have been studied un- der different viewpoints by Mori [Mor94, Mor95, Mor11] and by Bayer [Bay02], x Introduction Bayer and Travesa [BT07a, BT07b, BT08]. In particular, expansions for some modular and automorphic functions corresponding to the modular curve and also to the Shimura curve attached to a maximal order in the rational quater- nion algebra of discriminant D = 6 have been computed in [BT07a, BT07b], and the case of automorphic forms, also in the case D = 6, has been studied by Baba and Granath [BG12]. These have also been used to obtain expansions for automorphic forms using a numerical approach by Voight and Willis [VW14]. The computation of CM points in the case of triangle groups through numeric approximations is also considered by Voight [Voi06] and the determination of automorphic forms attached to Shimura curves through Jacquet-Langlands cor- respondence is considered by Voight in [Voi10]. Finally, the problem of giving equations for the genus 2 curves giving rise to the abelian surfaces corresponding to points in X(Γ) has been studied by Rotger in [Rot04] and a numerical approach has also been given by Bayer and Guàrdia [BG05]. In [BG08], Baba and Granath obtain a generating function (Hauptmodul) for the rational function field of the maximal Atkin-Lehner quo- tient of the Shimura curves attached to quaternion algebras of discriminants D = 6, 10 by means of Igusa invariants using the Hashimoto-Murabayashi fam- ilies [HM95], yielding therefore a formula which is similar in essence to the one relating j to the coefficients of a Weierstrass model of the corresponding elliptic curve. In [BT08] this Hauptmodul is related to the one computed in [BT07b]. Therefore, it is natural to consider whether we can make use of the expan- sions of the automorphic functions around CM points studied in [BT07b] to help us in the determination of an explicit uniformization of a genus 0 Atkin-Lehner quotient of a Shimura curve and contributing therefore to a better understanding of the open problems posed by Elkies [Elk98, §5.5], especially those concerning the computation of covers and CM points. This is the main problem we con- sider; in parallel, we will also deal with several other problems we will find in the way, as detailed next. In Chapter 1 we collect the basic concepts and results which will be required later. It is broken down into three sections, which deal with quaternion algebras and orders, the Shimura curve X(D,N) attached to an Eichler order of level N in a quaternion algebra of discriminant D and automorphic forms, respectively. In Chapter 2, we introduce the kroneckerian polynomials, which are the natural generalization to the case of Shimura curves of the classical modular equations. In order to do so, a complete study of the set of primitive elements in an Eichler order of a given norm coprime to the level of the order is car- ried out. Then, given a certain Atkin-Lehner quotient X(D,M)W of genus 0 xi of a Shimura curve X(D,M), a Hauptmodul t for the curve X(D,M)W over a certain number field K and a level N coprime to DM , the kroneckerian polyno- mial Ψ(X,Y ) = ΨΓ(D,M)￿W ￿,N,t(X,Y ) ∈ C[X,Y ] is defined. In Theorem 2.1, we prove that it is a symmetric (if N > 1), irreducible polynomial with coefficients in K of degree ψ(N) in each variable. Moreover, we see that it provides an equation for X(D,MN)W over K and that an equation for X(D,MN)￿W,ωN ￿ over K can be deduced from it. In Proposition 2.5, we study the factorization of Ψ(X,X) and we relate its zeros to the values of the Hauptmodul t at cer- tain CM points, similarly to the classical factoritzation of the modular equation Φ(X,X) in terms of class equations. Next, we study the singularities of the kro- neckerian polynomial of a certain level. We see that the set of singular points consists entirely of CM points and we provide a description of the number of branches through a point, which we simply call multiplicity, in Theorem 2.2. In Theorem 2.3, we focus on the case of prime power level, and we give a com- plete description of the set of singular points together with a formula for the multiplicity. In the second part of Chapter 2, we focus on those Shimura curves X(D,N) which have genus 0, which are the curves X(D, 1) with D = 6, 10, 22. In this case, in Theorem 2.4 we give a unified approach to the construction of Haupt- moduln for these three curves. All these Hauptmoduln are defined over quartic fields. However, we see that the corresponding kroneckerian polynomials are defined over Q and satisfy certain additional symmetry conditions correspond- ing to the action of the Atkin-Lehner involutions. Finally, in Theorem 2.5, for a kroneckerian polynomial of prime level attached to any of these Hauptmoduln, we study its reduction modulo p and deduce a formula, similar to that given by Kronecker congruence formula in the modular case, from Shimura reciprocity law. In Chapter 3, we begin by giving an algorithmic approach to the computation of fundamental domains for the groups Γ(D,N) based on Ford’s method. Then, we use it to compute fundamental domains for Γ(D, 1)￿WD￿ for D = 6, 10, 22. Next, we introduce the q-expansions and the differential equations satisfied by the Hauptmoduln together with their relation to linear differential equations, and use them together with the kroneckerian polynomials, to obtain an ex- plicit uniformization of the curve X(22, 1)W22 , which to our knowledge, was not known before. We recover as well uniformizations for the curves X(D, 1)WD for D = 6, 10 which had already been obtained by [Elk98] and [BT07b]. All these uniformizations are given in Theorem 3.1. Finally, we express our Hauptmodul for the curve X(22, 1)W22 in terms of Igusa invariants. In order to do so, we xii Introduction embed our curve in Igusa’s threefold and identify its image as a certain factor of an intersection of Humbert surfaces. This allows us to obtain an expression for our Hauptmodul in terms of Igusa invariants in Theorem 3.10 and to write an explicit family of genus 2 curves parametrized by automorphic functions in Q(X(22, 1)) giving rise to the corresponding abelian surfaces at each point (ex- cept for a finite set of exceptional points which is also computed). Finally the potential good reduction of the curves in the family is also studied. In Chapter 4 we begin by computing the whole set of couples (D,N) for which the curveX(D,N)WDN has genus 0, which is the set of curves for which we could follow the approach introduced in Chapter 3 in order to obtain an explicit uniformization. In Proposition 4.1, we give a list of all the 271 curves which have an Atkin-Lehner quotient of genus 0 together with all the information necessary to compute the genus and the number of elliptic cycles (and the corresponding quadratic orders) of any quotient of the curve X(D,N). Those with squarefree N had already been computed in [LMR06]. Moreover, we prove that all such curves X(D,N)WDN are isomorphic to P1Q. We can also readily see that the number of elliptic cycles of the curve X(D,N)WDN is bigger or equal than that of the curve X(D, 1)WD , which also has genus 0. Since the computational complexity of the approach given in Chapter 3 depends basically on the number of elliptic cycles, it is therefore better to obtain an explicit uniformization for the curve X(D, 1)WD and then use this one to obtain an uniformization for X(D,N)WDN . In this chapter, we will see how this can be done using again the kroneckerian polynomials introduced in Chapter 2 and the ideas used in [Elk98]. To illustrate the approach, we will consider the curves studied in Chapter 3, and we will uniformize the curves X(D,N), D = 6, 10, 22, together with all their quotients, for the 3 smallest possible values of N > 1 in each case. We will provide Hauptmoduln for all the rational quotients, and equations for all the remaining curves of genus 0, 1, 2 or hyperelliptic of higher genus. To our knowledge, explicit uniformizations for the cases (6, 11), (10, 9) and those of discriminant 22 had not been obtained before. In this case, the complexity of the computations depends on the degree of the cover X(D,N)WD → X(D, 1)WD , which is ψ(N) but, if N is composite (or a prime power), it can be broken down into smaller pieces. Moreover, we can work with numeric (or p-adic) approximations and we could even resort to some computer algebra software to deal with the easier cases. In Chapter 5 we study the algebraic properties of the coefficients of the q- expansions of automorphic forms attached to the groups Γ(D,N) around CM points. In order to do so, we introduce the type of q-expansions we are going xiii to consider and relate the coefficients to the values of certain operators, closely related to Shimura-Maaß differential operators. Then, following the ideas in [Shi75a], in Theorem 5.1 we obtain an algebraicity result for the coefficients of the q-expansions, when suitably normalized, formulated over precise num- ber fields. In Corollary 5.2, we consider the algebraicity of the coefficients of the q-expansions of automorphic functions attached to Γ(D,N), generalizing to an arbitrary D and over precise number fields the algebraicity result given in [BT08]. In Theorem 5.2 we prove the other implication, characterizing the field of definition of a form in terms of the coefficients of the q-expansion at a CM point. In Corollary 5.3, we obtain a different proof for a result by Ihara [Iha74], namely that the automorphic derivative of an automorphic function defined over a field K is as well an automorphic function defined over the same field. Finally, in Theorem 5.3, we deal with the action of the Galois group on the coefficients of the q-expansions, extending in this way the Shimura reciprocity law, which deals with the action of the Galois group on the values of the automorphic functions, to the whole set of coefficients. In Chapter 6 we deal with some of the computational applications of the tools introduced in the previous chapters. It is broken down into four differ- ent sections. In Section 6.1 we study the explicit computation of kroneckerian polynomials of a certain level when q-expansions around certain CM points are known and we give some examples in the case D = 6. More polynomials have been computed in order to deal with the later applications, more precisely for D = 6, 10, 22 we have computed the kroneckerian polynomials of prime level for all primes not dividing D up to 41. In Section 6.2 we deal with the problem of computing the values of the Hauptmodul at a given CM point in a determin- istic way without making use of approximations by using Proposition 2.5. In particular, we apply this to the computation of the rational and quadratic CM points of the curves X(D, 1)WD , obtaining in particular a new proof that the rational values given by Elkies in [Elk98] for the cases D = 6, 10 are correct (we give as well an omitted value in the case D = 10). A different proof following a completely different method has been given by Errthum [Err11]. Our approach can also be used to check the correctness of the tables given in [Elk98] for the cases D = 14, 15. We do not restrict ourselves to the case of the values of the Hauptmoduln and we provide a method to obtain as well the first few terms of a q-expansion at a given CM point. After that, more terms can be computed using the differential equation satisfied by the Haupmodul. We provide as well tables for the first few terms of the q-expansions around all the rational CM points. However, to make these expansions really functional, the local parameter with xiv Introduction respect to which we consider the expansion has to be computed and this is what we do in Section 6.3. Note that the results in this section, with the exception of three values in the case D = 6, are obtained by approximation and recognition and, even though we are confident that they are correct, we know no way to prove it and therefore we can only be certain that they are good approximations (at least up to 150 digital digits, but more digits can be checked) of the real values. In any case, this suffices in general to obtain numeric approximations for the value of the function or its derivative in a neighborhood of the point. Finally, in Section 6.4, we deal with the problem of obtaining q-expansions of automorphic forms. In order to do so, we assume that X(D,N) is such that X(D,N)WDN has genus 0 and we compute a basis for the space of weight k au- tomorphic forms corresponding to a curve X(D,N) in terms of a Hauptmodul for X(D,N)WDN . We study as well how to compute the action of a certain Hecke operator in the space of automorphic forms of a certain weight. Then, we provide a set of generators and relations for the graded algebra of automorphic forms for D = 6, 10, 22 and the levels for which we computed uniformizations in the previous chapters. Moreover, some examples of q-expansions are also provided. Finally, we must add that almost all the computations in this thesis have been carried out using Magma [BCP97] or Mathematica [Wol08], but at some point we have also used Sage [Dev12], PARI [The14] and Macaulay2 [GS]. Acknowledgments I cannot finish without thanking all the people who, at some point and in some way, have contributed to make this thesis possible. In first place I want to express my gratitude to Artur Travesa for introducing me to this wonderful area of mathematics and for always having time to listen to my mathematical ideas, even in difficult times. I also want to thank Pilar Bayer for her useful advices but also for her enthusiasm and motivation. I also feel indebted to Luis Dieulefait, for guiding me through some of my introductory research, because I learned much more than just number theory. I wish to thank as well all my (current and former) colleagues at the Depar- tament d’Àlgebra i Geometria. And of course I want to extend this gratitude to all the members of the Seminari de Teoria de Nombres de Barcelona of the xv Universitat de Barcelona, the Universitat Autònoma de Barcelona and the Uni- versitat Politècnica de Catalunya, for their motivation and to help me make a more complete picture of what number theory is. I also thank Alexei Skorobogatov, Kevin Buzzard and the Department of Mathematics at Imperial College London for their hospitality during my stay there. Finalment, aquesta tesi no hauria estat possible sense tota aquella gent que sempre està i ha estat al meu costat, a Barcelona i també a Breda, els meus amics i la meva família. Moltes gràcies a tots. Chapter 1 Quaternion algebras, Shimura curves and automorphic forms In this chapter we are going to introduce the basic notions and some well-known results which will be required as a background. We will break this chapter into three sections. The first one deals with quaternion algebras, quaternion orders and some other related notions. Based on this, we introduce Shimura curves and some of their properties in Section 1.2 and automorphic forms attached to groups of quaternion units in Section 1.3. Some references for the results in this chapter are [Vig80], [Shi71], [Ogg83], [AB04]. 1.1 Quaternion algebras Let K be a field of characteristic different from 2. Definition. A quaternion K-algebra H is an unitary and associative K-algebra such that: (a) dimK H = 4; (b) it is central, i.e. its center is K, Z(H) = K; (c) it is simple, i.e.H has no non-trivial bilateral ideals. 1 2 Quaternion algebras, Shimura curves and automorphic forms We denote by H∗ its group of units, i.e. H∗ = {h ∈ H : ∃h￿ ∈ H, h￿h = hh￿ = 1}. As in the case of generalK-algebras, a morphism of quaternionK-algebras is aK-linear morphism of unitary rings and we can naturally considerK embedded into H. Given an element k ∈ H∗, we can consider an automorphism of H given by h ￿→ khk−1 for every h ∈ H. These automorphisms of H are called inner automorphisms. The well-known Theorem of Skolem-Noether states that any automorphism of H is inner. In our case, since the characteristic ofK is not 2, every quaternionK-algebra H admits the following description: there exists a K-basis {1, i, j, ij} of H such that i2 = a, j2 = b and ij = −ji for some a, b ∈ K∗. Conversely, aK-basis with the previous properties, and 1 as unity, together with the associativity, defines a quaternion K-algebra. This algebra will be denoted, as usual, by ￿ a,b K ￿ . Remark. Note that different choices of a, b ∈ K∗ can yield isomorphic quater- nion K-algebras. For example, ￿ a,b K ￿ ∼= ￿ b,aK ￿. A quaternion K-algebra H = ￿ a,b K ￿ is endowed with an involutive anti- automorphism called conjugation which is denoted by w ￿→ w. If we consider the natural K-basis on H and we write w = x+ yi+ zj+ tij with x, y, z, t ∈ K, then w = x− yi− zj − tij. Now that the conjugation has been introduced, we can immediately define the (reduced) trace tr(w) = w + w and the (reduced) norm nr(w) = ww. Using the previous expression of w in the basis {1, i, j, ij}, one has that tr(w) = 2x and nr(w) = x2 − ay2 − bz2 + abt2. The group of units H∗ is exactly the set of elements with non-zero norm. Remark. Since we will not use the general notions of trace and norm for K- algebras (cf. [Bou58]), whenever we refer to traces and norms in quaternion algebras, it will be implied that they are the reduced traces and norms above. Given an integral domain R, we will denote byM(2, R) the ring of 2×2 ma- trices over R, by GL(2, R) the corresponding linear group of invertible matrices inM(2, R), by SL(2, R) the corresponding special linear group of determinant 1 matrices and by PGL(2, R) = GL(2, R)/R∗, PSL(2, R) = SL(2, R)/{±1} the corresponding projective groups. 1.1 Quaternion algebras 3 Example. If H =M(2,K) and w = ( x yz t ), then w = ￿ t −y −z x ￿ is the transposed adjoint matrix of w. In particular, the reduced trace and norm coincide with the trace and determinant of the matrix. If a quaternion K-algebra is isomorphic toM(2,K), it will simply be called a matrix K-algebra. If that is not the case, then the quaternion K-algebra is a skew field and it will be called a division K-algebra. From now on we will assume that K is a number field and we will denote its completion at a place ν by Kν . We will write OK and OK,ν their respective rings of integers. Definition. Let H be a quaternion K-algebra. For each place ν of K, let Hν = Kν ⊗K H be the corresponding quaternion Kν-algebra. If Hν is a matrix algebra, we will say that H is unramified (or does not ramify) at ν. Otherwise, if Hν is a division algebra, we will say that H is ramified (or ramifies) at ν. Given H = ￿ a,b K ￿ and a place ν of K, the Hasse invariant at ν is defined as ￿ ￿ a, b K ￿ ν = ￿ 1, if ν does not ramify in H, −1, if ν ramifies in H. In the particular case K = Q, the Hasse invariant at p, ￿ ￿ a,b Q ￿ p , coincides with the Hilbert symbol (a, b)p, and therefore it admits an explicit description in terms of Legendre symbols (cf. [Ser73]). Now we can state the well-known classification theorem. Theorem 1.1. (1) A quaternion K-algebra H ramifies at a finite and even number of places. (2) Two quaternion K-algebras are isomorphic if and only if they ramify at the same places. (3) Given an even number of non-complex places of K, there exists a quaternion K-algebra which ramifies at exactly these places. Definition. The (reduced) discriminant DH of a quaternion K-algebra H is the integral ideal of OK which is equal to the product of the prime ideals that ramify in H. 4 Quaternion algebras, Shimura curves and automorphic forms Therefore, two quaternion K-algebras are isomorphic if and only if they have the same discriminant. In particular, a quaternionK-algebra is a matrix algebra if and only if it has discriminant OK . Another notion playing an important role is that of splitting fields. Definition. Let H be a quaternion K-algebra and F a field extension of K. The field F splits H if and only if F ⊗K H is a matrix F -algebra. As we will see later on, among the fields splitting a quaternion K-algebra, those which are quadratic extensions of K will be of special interest, and there- fore the following criterion will be particularly useful. Theorem 1.2. Let H be a quaternion K-algebra and F |K a quadratic extension. Then, the following statements are equivalent: (i) F splits H. (ii) F is isomorphic as a K-algebra to a maximal subfield of H containing K. (iii) There exists an embedding of K-algebras F ￿→ H. (iv) Every place ν in K which ramifies in H is not totally split in F . The case we will be interested in is the case K = Q, that is, the case of rational quaternion algebras. Definition. Let H be a rational quaternion algebra. If R splits H, we say that H is indefinite. Otherwise, we say that H is definite. Remark. Given a rational quaternion algebra H = ￿ a,b Q ￿ , H is indefinite if and only if at least one of a or b is positive. In the case of rational quaternion algebras, since the discriminant is an ideal of Z, we can identify it with a positive integer generating this ideal. The definite or indefinite character of a rational quaternion algebra H can then be read from the number of prime factors of DH, namely an odd number of prime factors corresponds to the definite case and an even number to the indefinite one. On the other hand, the indefinite character of H is equivalent to the existence of an embedding H ￿→M(2,R). Let H = ￿ a,b Q ￿ be an indefinite rational quaternion algebra. Permuting a and b, if necessary, we can assume that a > 0. Then Φ : H ￿→M(2,R) defined 1.1 Quaternion algebras 5 by Φ(x+ yi+ zj + tij) = ￿ x+ y √ a z + t √ a b(z − t√a) x− y√a ￿ is an embedding. 1.1.1 Quaternion orders Let R be a Dedekind domain and K its fraction field. Let H be a quaternion K-algebra. Definition. An element h ∈ H is said to be integral over K if both tr(h) and nr(h) lie in R. In contrast to the abelian case, the set of integral elements in a quaternion algebra is not a ring except in some particular cases. For this reason the notion of order is introduced. First of all we consider the notion of ideal. Definition. An R-lattice Λ of H is a finitely generated R-module contained in H. An R-ideal I of H is an R-lattice such that K · I = H. The inverse of an R-ideal I is the R-ideal I−1 = {h ∈ H : IhI ⊂ I}. An R-ideal I is integral if all its elements are integral. Definition. A subsetO ofH is anR-order if it satisfies any of the three following equivalent conditions: (a) O is a subring of H which is a free R-module of rank 4. (b) O is an R-ideal which is a ring. (c) O is a ring containingR, whose elements are integral and such thatKO = H. Given an R-ideal I in H, its associated right and left orders are defined as follows: Or(I) = {h ∈ H : Ih ⊆ I} and Ol(I) = {h ∈ H : hI ⊆ I}. The R-ideal I is integral if and only if it is contained in the associated orders. Then I is a right (resp. left) ideal of Or(I) (resp.Ol(I)). We say that I is bilateral if Or(I) = Ol(I). The R-ideal I is principal if there exists h ∈ H such that I = hOr(I) = Ol(I)h. 6 Quaternion algebras, Shimura curves and automorphic forms Definition. Given an R-order O, its different DO is the bilateral R-ideal of H defined as the inverse of the codifferent, DO−1 = {α ∈ H : tr(αO) ⊆ R}. The (reduced) discriminant DO of O is the reduced norm of its different DO, namely, the fractional ideal of R generated by the norms of the elements in DO. The discriminant of a given R-order can be computed easily using the fol- lowing result. Proposition 1.1. Let O be an R-order in H. (1) If {v1, v2, v3, v4} is an R-basis of O, then D2O = R det(tr(vivj)). (2) Let O ⊆ O￿ be R-orders in H. Then DO￿ divides DO, and the two are equal if and only if the orders are equal. (3) Given ψ : H→ H an automorphism of H, then Dψ(O) = DO. In any quaternion K-algebra, every R-order is contained in a maximal one. Maximal orders will play an important role as soon as we introduce Shimura curves. However, we will consider them in the bigger family of Eichler orders. Definition. An Eichler R-order in a quaternion K-algebra H is the R-order obtained as the intersection of two maximal R-orders in H. Now, we are going to study in more detail the properties and structure of these orders. First of all, it is obvious that the class of Eichler orders is invariant under isomorphisms. In particular, the set of Eichler orders of a quaternion algebra is closed under conjugation. What is not so obvious is when two such orders are indeed conjugated. We will split the study of Eichler orders in two cases: we will begin by the local Eichler orders and then we will move on to the global ones. Local Eichler orders Let ν be a finite place of a number field K and let Kν the corresponding com- pletion. Let Hν = Kν ⊗K H. Let Rν be the ring of integers of Kν and let ￿ be a uniformizer of Rν . Now two possible cases appear depending on whether Hν is a division algebra or a matrix algebra. For this reason, we will summarize the classification of Eichler orders in two different theorems. 1.1 Quaternion algebras 7 Theorem 1.3. Let Hν be a local division Kν-algebra. Then, Oν = {h ∈ Hν : nr(h) ∈ Rν} is the unique maximal order. Hence, Oν is the unique Eichler Rν-order in Hν . Moreover, the ideal ￿Rν ramifies in Oν ; more precisely, ￿Oν = m2, where m is the unique maximal ideal of Oν . Theorem 1.4. Let Oν ⊂ M(2,Kν) be an Rν-order. Then, the following are equivalent: (i) Oν is an Eichler order. (ii) There exists a unique pair of maximal orders {O1,O2} in M(2,Kν) such that Oν = O1 ∩O2. (iii) There exists an integer n ∈ N such that Oν is conjugated to the order￿ Rν Rν ￿nRν Rν ￿ . Moreover, n is uniquely determined. (iv) The order Oν contains a subring conjugated to￿ Rν 0 0 Rν ￿ . The ideal NOν = ￿nRν is called the level of the local Eichler order Oν . Therefore, we can now define the general notion of level of a local Eichler order. Definition. Let Oν be a local Eichler order in a quaternion Kν-algebra Hν . The level of Oν is the ideal N(Oν) = ￿ Rν , if Hν is a division algebra, NOν , if Hν is a matrix algebra. Remark. For any local Eichler order Oν , DOν = N(Oν). 8 Quaternion algebras, Shimura curves and automorphic forms Global Eichler orders Let H be a quaternion algebra over a number field K with ring of integers R and let O be an R-order in H. For any place ν of K, let Oν = Rν ⊗R O. For a finite place ν, Oν is a local Rν-order and for an infinite place ν, we consider Rν = Kν and then Oν = Hν . In particular, considering the natural inclusions H ￿→ Hν , we obtain that O = ￿ ν(H ∩ Oν). Moreover, it can be proved that (DO)ν = DOν . Proposition 1.2. O is a maximal order if and only if Oν is an Rν-maximal order for every finite place ν. Moreover, the order O is maximal if and only if DO = DH. Observe that the previous result gives a simple procedure to check whether a given order is maximal or not. The local character of the maximality is also true for Eichler orders. Proposition 1.3. O is an Eichler R-order if and only if Oν is a local Eichler Rν-order for every finite place ν. Thus, this proposition provides a very useful way to recognize Eichler orders using the local characterization we have seen before. Now, since we have a notion of level for local Eichler orders, it can be shown that we can pack them together in order to define the level of a global Eichler order. Definition. The level NO of the global Eichler order O is the unique ideal N in R such that Nν = Rν ⊗R N is the level of the local Eichler Rν-order Oν . Eichler orders are not characterized in general in terms of their discriminant, as it was the case for maximal orders. However, some conditions and particular cases can still be obtained. Proposition 1.4. Let O be an order in a rational quaternion algebra H. (1) The level of an Eichler order in H is coprime to DH. (2) If DO = DHN with N a squarefree integer, then O is an Eichler order of level N . Also in the case K = Q, we will denote by O(D,N) an Eichler order of level N in the algebra of discriminant D. In this situation, we can give an equivalent interpretation of the notion of level. 1.1 Quaternion algebras 9 Proposition 1.5. Let O(D,N) ⊆ O(D, 1). Then, the index of O(D,N) in O(D, 1) as Z-modules is N . Finally, we recall some results regarding the existence and the unicity of Eichler orders of a fixed level modulo conjugation. Proposition 1.6. Let Λ be a lattice in a quaternion K-algebra H. For each finite place ν of K, let Lν be a local lattice in the local quaternion Kν-algebra Hν . Assume that Lν = Λν for all but a finite number of ν. Then, there exists a lattice Λ￿ in H such that Λ￿ν = Lν for all ν. Corollary 1.1. Let H be a quaternion Q-algebra of discriminant D. Then, for every N coprime to D there exist Eichler orders of level N . Theorem 1.5. Let H be an indefinite quaternion Q-algebra. Then, there is only one conjugacy class of Eichler orders having the same level and all (left or right) ideals of an Eichler order in H are principal. Bilateral ideals and normalizer of Eichler orders Given an order O in H, its normalizer is Nor(O) = {σ ∈ H∗ : σOσ−1 = O}. Let O = O(D,N) be an Eichler order of level N in a rational quaternion algebra of discriminant D. For any m￿DN , i.e.m|DN and gcd(m,DN/m) = 1, there exists an ideal I = I(m) such that I2 = mO, which can be described locally as follows: • If p ￿ m, Ip = Op. • If p|m and p|D, Ip = Op ￿O∗p. • If pk￿m and pk￿N and we identify Op with￿￿ a b cpk d ￿ : a, b, c, d ∈ Zp ￿ , then Ip is identified with ￿ 0 1 pk 0 ￿ Op = Op ￿ 0 1 pk 0 ￿ . 10 Quaternion algebras, Shimura curves and automorphic forms Since all ideals in O are principal, we can write I(m) = wmO = Owm, for a certain wm ∈ O of norm m. In particular, O = wmOw−1m . Theorem 1.6. Let O = O(D,N) be an Eichler order of level N in a rational quaternion algebra of discriminant D. Then, O∗\Nor(O) ∼= (Z/2Z)r, where r is the number of distinct primes dividing DN . The elements wm above provide representatives for the non-trivial classes. Embeddings of quadratic orders Definition. Consider H an indefinite quaternion Q-algebra and F a quadratic field. Consider as well a quaternion order O ⊂ H and a quadratic order Λ ⊂ F . An embedding of Λ into O is an embedding φ : F ￿→ H such that φ(Λ) ⊂ O. It is called optimal if φ(F ) ∩O = φ(Λ). We will denote E(O,Λ) and E∗(O,Λ) the set of embeddings and optimal embeddings respectively. Given a subgroup G ⊂ Nor(O), we can consider the action E(O,Λ)×G → E(O,Λ) (φ,σ) ￿→ φσ, where φσ(α) = σ−1φ(α)σ for all α ∈ F . This action restricts to E∗(O,Λ) and therefore induces equivalence relations on both sets. We denote by ν(O,Λ;G) the number of equivalence classes of optimal embeddings modulo G. The fol- lowing theorem gives an explicit formula for this number when G = O∗. Theorem 1.7. Let O = O(D,N) be an Eichler order of level N in an indefinite quaternion Q-algebra of discriminant D. Let Λ(d,m) be the quadratic order of conductor m in Q( √ d). Then, ν(O,Λ(d,m);O∗) = h(d,m) ￿ p|DN νp(O,Λ(d,m);O∗), where h(d,m) denotes the ideal class number of the order Λ(d,m), ψ denotes the Euler ψ function and 1.2 Shimura curves 11 • If p|D, νp(O,Λ(d,m);O∗) = ￿ 1− ￿ DF p ￿ , if p ￿ m, 0, otherwise. • If p￿N , νp(O,Λ(d,m);O∗) = ￿ 1 + ￿ DF p ￿ , if p ￿ m, 2, otherwise. • Assume N = pru1, p ￿ u1, r ≥ 2, and m = pku2, p ￿ u2: – If r ≥ 2k + 2, νp(O,Λ(d,m);O∗) = ￿ 2ψ(pk), if ￿ DF p ￿ = 1, 0, otherwise . – If r = 2k + 1, νp(O,Λ(d,m);O∗) =  2ψ(pk), if ￿ DF p ￿ = 1, pk, if ￿ DF p ￿ = 0, 0, otherwise. – If r = 2k, νp(O,Λ(d,m);O∗) = pk−1 ￿ p+ 1 + ￿ DF p ￿￿ . – If r ≤ 2k − 1, νp(O,Λ(d,m);O∗) = ￿ pk/2−1(p+ 1), if k is even, 2p(k−1)/2, if k is odd. 1.2 Shimura curves 1.2.1 The upper half-plane Let H = {z = x + iy ∈ C : y > 0} be the Poincaré upper half-plane endowed with the structure provided by the hyperbolic metric, 1 y2 ((dx)2 + (dy)2). The hyperbolic distance between a couple of points z, w ∈ H is given by dH(z, w) = arccosh ￿ 1 + |z − w|2 2Im(z)Im(w) ￿ = 2arctanh ￿￿￿￿￿z − wz − w ￿￿￿￿￿ . Then, the set of hyperbolic lines are the semilines which are parallel to the imaginary axis and the semicircles whose center lies on the real axis. Moreover, 12 Quaternion algebras, Shimura curves and automorphic forms the hyperbolic circle of center z = x+ iy and radius r is the euclidean circle of center x+ icosh(r)y and radius sinh(r)y. A hyperbolic polygon is a closed subset of H bounded by a simple curve formed by a finite number of segments of hyperbolic lines. The hyperbolic volume, with respect to the corresponding measure dw = dxdy y2 , of a hyperbolic polygon can be computed easily by means of the following result. Theorem 1.8 (Lambert). Let P be a hyperbolic polygon with vertices {vi}ni=1 and internal angle θi at the vertex vi, i = 1, . . . , n. Then, the hyperbolic volume of P is VH(P) = (n− 2)π − ￿n i=1 θi. 1.2.2 Homographic transformations An homographic transformation γ : H→ H is a transformation given by γ(z) = az + b cz + d , a, b, c, d ∈ R, ad− bc > 0. This set of transformations with the composition is a group isomorphic to PSL(2,R) by means of ￿ a b c d ￿ ￿→ az + b cz + d . Therefore, we have a faithful action of PSL(2,R) on H, which gives rise to an action on H of GL+(2,R) ⊂ GL(2,R), the subgroup of matrices of positive determinant. When there is no risk of con- fusion, we will write γ for both the transformation and a representative matrix. More generally, given ￿ a b c d ￿ ∈ PGL(2,R), we define the corresponding trans- formation as γ(z) = az + b cz + d , if ad − bc > 0, and γ(z) = az + b cz + d , if ad − bc < 0. Observe that this definition does not depend on the representative. Definition. Let γ be an homographic transformation different from id. (1) γ is elliptic if it has a fixed point z ∈ H. (2) γ is hyperbolic if it has two fixed points in R ∪ {∞}. (3) γ is parabolic if it has a unique fixed point in R ∪ {∞}. 1.2 Shimura curves 13 Let Γ ⊂ PSL(2,R) be a discrete subgroup (it is usually called a fuchsian group) and consider the natural action of Γ on the upper half-plane, which is properly discontinuous. This group gives rise therefore to an equivalence relation on the upper half-plane, namely, z, z￿ ∈ H are equivalent if and only if there exists γ ∈ Γ such that γ(z) = z￿. A point z ∈ H ∪ R ∪ {∞} is called: (1) elliptic by Γ if there exists γ ∈ Γ elliptic such that γ(z) = z; in particular, z ∈ H. In this case we denote by Γz = {γ ∈ Γ : γ(z) = z} its isotropy group and we define the order of the elliptic point as the order of its isotropy group; (2) hyperbolic by Γ if there exists γ ∈ Γ hyperbolic such that γ(z) = z; in particular, z ∈ R ∪ {∞}; (3) parabolic (or a cusp) by Γ if there exists γ ∈ Γ parabolic such that γ(z) = z; in particular, z ∈ R ∪ {∞}. 1.2.3 Fundamental domains Let Γ ⊂ PSL(2,R) be a discrete subgroup and let H∗ be the union of H and the cusps of Γ. Assume further that Γ is cocompact, i.e.Γ\H∗ is compact. Definition. A convex and closed hyperbolic polygon D ⊂ H∗ will be called a fundamental domain for Γ if the following two conditions are satisfied: (a) For all z ∈ H, there exists γ ∈ Γ such that γz ∈ D. (b) If there exist z ∈ D and γ ∈ Γ￿{1} such that γz ∈ D, then z ∈ D￿ int(D). We will see the existence of a fundamental domain for Γ, together with an explicit description of the corresponding polygon, in Theorem 1.9. However, we will not only consider the previous set, but we will consider as well the action of the group on it. Given a fundamental domain D for a group Γ, the action of Γ induces a pairing between the sides of D (it may be necessary to split some sides by adding additional vertices, for example if a side is identified with itself). Whenever we refer to a fundamental domain we will associate to it a pairing between its sides and the corresponding identification between its vertices. 14 Quaternion algebras, Shimura curves and automorphic forms A cycle of a fundamental domain is an orbit of vertices under the action of Γ. A cycle is called elliptic of order k if it consists of elliptic vertices of order k. A cycle is called parabolic if it consists of parabolic vertices (we say that its order is ∞). A cycle is called accidental if it is neither elliptic nor parabolic. The length of a cycle is the number of vertices in it. Remark. Since the property of being elliptic of a given order (or parabolic) is invariant under the action of Γ, every cycle is in one of the cases above. The number of elliptic and parabolic cycles is an intrinsic property of the group, but the number of accidental cycles is not. Moreover, if we consider the transformations identifying sides together with the relations provided by the cycles, we obtain a presentation of the group. There are various methods to compute fundamental domains for the group Γ, cf. [For29], [Leh64], [AB04], [Kat92]. We will follow Ford’s method and in order to make it work in all situations we will need to move to the slightly more general setting of groups conjugated to fuchsian groups inside PSL(2,C). Definition. For a, b, c, d ∈ C, ad− bc = 1, consider the homographic transfor- mation γ : P1C = C ∪ {∞} → P1C = C ∪ {∞}, z ￿→ γ(z) = az+bcz+d . Assume further that c ￿= 0, or equivalently γ(∞) ￿= ∞. Then we define the isometric circle attached to γ as Cγ = {z ∈ C : |cz + d| ≤ 1}. We denote as well by Eγ = P1C ￿ Cγ the exterior of the isometric circle. Theorem 1.9 ([For29], [Leh64]). Let z0 ∈ H be a non-elliptic point for Γ and let γ0 ∈ PSL(2,C) be the complex homographic involution which permutes z0 and ∞. Then, γ0 ￿ ￿ γ∈Γ Eγ0γγ0 ￿ ∩H, where · denotes the complex topological closure, is a fundamental domain for Γ. Some examples and more details on the computation will be given in Chap- ter 3. 1.2 Shimura curves 15 1.2.4 Groups of quaternion units and the corresponding Riemann surfaces Now we are going to use the quaternion orders previously introduced to obtain the groups Γ we will be working with. Let H = ￿ a,b Q ￿ be a rational indefinite quaternion algebra of discriminant D and fix Φ : H ￿→ M(2,R). Let also N be a natural number such that gcd(D,N) = 1. Let O(D,N) be an Eichler order in H of level N . Consider now O(D,N)1 = {α ∈ O(D,N) : nr(α) = 1}. This is a group of index 2 in O(D,N)∗. Observe, moreover, that it is defined byD and N up to isomorphism. Now, we define Γ(D,N) = Φ(O(D,N)1)/{±1} ⊂ PSL(2,R). Γ(D,N) is then a cocompact fuchsian group and its action on the upper half-plane provides a quotient set Γ(D,N)\H∗, which naturally inherits an analytical structure from that of H, after suitably modifying it at the elliptic and parabolic points (cf. [Shi71]). Therefore, Γ(D,N)\H∗ acquires a natural structure of compact Riemann surface and the fundamental domain together with the identifications provides a plane model of it. Fix Γ(D,N) and a fundamental domain D(D,N) for it. Let ei(D,N) denote the number of elliptic cycles of order i, e∞ the number of parabolic cycles, V (D,N) = VH(D(D,N))/(2π) the normalized volume and g(D,N) the genus of Γ(D,N)\H∗. Theorem 1.10. Consider the group Γ(D,N). Then, only elliptic points of orders 2 and 3 can occur and cusps exist if and only if D = 1. Moreover, we have the following formulas for all these invariants: (1) e2(D,N) =  ￿ p|D ￿ 1− ￿ −4 p ￿￿￿ p|N ￿ 1 + ￿ −4 p ￿￿ , if 4 ￿ N, 0, otherwise. (2) e3(D,N) =  ￿ p|D ￿ 1− ￿ −3 p ￿￿￿ p|N ￿ 1 + ￿ −3 p ￿￿ , if 9 ￿ N, 0, otherwise. (3) e∞(1, N) = ￿ d|N d>0 φ(gcd(d,N/d)), where φ is the Euler φ-function. 16 Quaternion algebras, Shimura curves and automorphic forms (4) V (D,N) = N6 ￿ p|D (p− 1) ￿ p|N (1 + 1p ). (5) 2− 2g(D,N) = −V (D,N) + 12e2(D,N) + 23e3(D,N) + e∞(D,N). Remark. The formula relating the genus, the volume and the number of elliptic cycles is a particular case of a formula which is valid for more general fuchsian groups, Γ, such that VH(Γ\H∗) < ∞. More precisely, if we denote by e∞(Γ) the number of parabolic cycles by Γ and ei(Γ) the number of elliptic cycles of order i by Γ, then 1 2π VH(Γ\H∗) = 2g(Γ\H∗)− 2 + e∞(Γ) + ￿ i i−1 i ei(Γ). 1.2.5 Complex Multiplication (CM) points Consider O(D,N) an Eichler order of level N in an indefinite quaternion algebra H of discriminant D together with an embedding Φ : H ￿→M(2,R). Definition. Let F be an imaginary quadratic field and Λ ⊂ F be an order such that E∗(O(D,N),Λ) ￿= ∅. Let φ ∈ E∗(O(D,N),Λ) and consider zφ ∈ H the common fixed point by all the transformations in Φ(φ(F ∗)). We say that zφ is a complex multiplication (CM) point by Λ. It is worth noting that, if z ∈ H is fixed by the transformation attached to γ = Φ(h), for h ∈ O(D,N)￿ Z, then z is a CM point by a certain order in the field Q( √ d), with d = tr(γ)2 − 4nr(γ) < 0. A very particular case of CM points are the elliptic ones. Namely, z ∈ H is elliptic of order 2 (resp. 3) if and only if it is a CM point by the ring of integers of Q(i) (resp.Q( √−3)). Moreover, the set of CM points by a certain order Λ is invariant under the action of Γ(D,N), therefore we can consider CM(D,N, d,m) the set of CM points in Γ(D,N)\H by the order Λ of conductor m in the quadratic field of discriminant d < 0. If D,N are clear from the context, then we will simply write CM(Λ) = CM(D,N, d,m). We will denote cm(D,N, d,m) = #CM(D,N, d,m). Theorem 1.11. Given an Eichler order O(D,N), cm(D,N, d,m) = ν(O(D,N),Λ(d,m);O(D,N)∗), as introduced in Theorem 1.7. 1.2 Shimura curves 17 We refer to [AB04] for the details on the proof of the statements above and on the computation of CM points. 1.2.6 Ring Class Fields Let K be an imaginary quadratic field, OK its ring of integers and m ⊂ OK an ideal of K. Definition. Let I(OK ,m) be the group of all fractional ideals in K coprime to m and let P1(OK ,m) be the subgroup of I(OK ,m) generated by the principal ideals αOK , with α ≡ 1 (mod m). Then, we define the ideal class group of modulus m as C(OK ,m) = I(OK ,m)/P1(OK ,m). Note that for m = (1), it is just the classical ideal class group. As in this case, for the general one we have as well the following result. Proposition 1.7. The group C(OK ,m) is finite. A similar construction to the one above is given by the ideal class groups corresponding to quadratic orders. Definition. Let Λ ⊂ OK be the order of conductor f . The ideal class group of Λ, C(Λ), is defined as follows C(Λ) = I(OK , f)/PZ(OK , f), where I(OK , f) denotes the subgroup of the group of fractional ideals in K gen- erated by the integral ideals of norm prime to f and PZ(OK , f) is the subgroup generated by the principal ideals of the form αOK , with α ≡ a (mod fOK) for some integer a coprime to f . We denote by h(Λ) its order. Remark. This group can be rewritten in terms of proper ideals in Λ, but the presentation given above will be enough for our purposes. Theorem 1.12. Let Λ be the order of conductor f in K. Then, h(Λ) = h(OK)f [O∗K : Λ∗] ￿ p|f ￿ 1− ￿ DK p ￿ 1 p ￿ , where DK stands for the discriminant of K and ￿ DK · ￿ denotes the Kronecker symbol. 18 Quaternion algebras, Shimura curves and automorphic forms Proposition 1.8. Let L|K be a finite abelian field extension, p a prime of K unramified in L and P a prime of L over p. Then, there exists a unique element￿ L|K p ￿ ∈ Gal(L|K) such that for any a ∈ OL,￿ L|K p ￿ (a) ≡ aNorm(p) (mod P). Moreover, this element is independent of the choice of P. Definition. Let L|K be a finite abelian extension and m an ideal of K which is divisible by all ramified primes in L. Given a = r￿ i=1 prii , ri ∈ Z, for certain unramified primes pi ￿ m. We define the Artin symbol as￿ L|K a ￿ = r￿ i=1 ￿ L|K pi ￿ri . Theorem 1.13. Let L|K be a finite abelian extension and m an ideal of K which is divisible by all ramified primes in L. (1) The map Φm : I(OK ,m) → Gal(L|K), a ￿→ ￿ L|K a ￿ is a surjective morphism. (2) If the exponents of the primes in m are large enough, P1(OK ,m) ⊂ ker(Φm). The ideal m for which the second part of the previous theorem holds is not unique, but can be chosen in a minimal way according to the following theorem. Theorem 1.14. Given L|K a finite abelian extension, there exists f = f(L|K) such that: 1.2 Shimura curves 19 (1) A prime of K ramifies in L if and only if it divides f. (2) If m is an ideal which is divisible by all the primes of K ramified in L, then P1(OK ,m) ⊂ ker(Φm) if and only if f|m. f is called the conductor of L|K. The last central result of the theory is that we can go the other way around. Theorem 1.15. Let m be an ideal of OK and let H be a subgroup of I(OK ,m) such that P1(OK ,m) ⊂ H ⊂ I(OK ,m). Then, there exists a unique abelian extension L|K such that all ramified primes divide m and H = ker(Φm). Therefore, we have a way to attach finite abelian extensions of K to sub- groups H of the form P1(OK ,m) ⊂ H ⊂ I(OK ,m). Definition. Given an ideal m, we define the ray class field attached to m as the finite abelian extension of K, Km|K, corresponding to H = P1(OK ,m). Its Galois group is isomorphic to C(OK ,m). If m = (1), it is called the Hilbert Class Field and its Galois group is isomorphic to the ideal class group. Definition. Given an order Λ ⊂ OK of conductor f , we have P1(OK , f) ⊂ PZ(OK , f) ⊂ I(OK , f), and therefore there exists an abelian extension of K, KΛ|K, contained in Kf whose Galois group is isomorphic to C(Λ). KΛ|K is called the Ring Class Field attached to Λ. Note that Theorem 1.12 provides a formula for the degree of the Ring Class Field. This theory, which has been presented up to now in terms of ideals, can be restated in terms of ideles. At some point it will be useful to have this interpretation and therefore we are going to introduce it now. We will keep talking about imaginary quadratic fields for coherence. 20 Quaternion algebras, Shimura curves and automorphic forms Definition. An adele α of K is a family α = (αp)p, with αp ∈ Kp, indexed by the primes p of K, such that αp is integral except for a finite number of p. We denote by AK the set of adeles of K, which has structure of ring by the operations induced componentwise. We denote by IK = A∗K the ideles of K. The group CK = IK/K∗ is called the idele class group. Given an ideal m = ￿n i=1 p ni i , we define ImK = {α = (αp) ∈ IK : αp ∈ OKp ,αpi ∈ 1 + pnii }. We define as well CmK = ImKK∗/K∗. Theorem 1.16. The morphism κ : IK → I(OK , 1), α ￿→ ￿p pνp(αp), induces an isomorphism κm : CK/C m K → C(OK ,m). Therefore, given a ∈ CK we can attach to it Artin symbols. In the adelic notation, they are denoted (a, L|K). 1.2.7 The Shimura curve X(D,N) Let D,N be positive integers. Assume that D is the product of an even number of primes and that N is coprime to D. Now consider an indefinite quaternion algebra H over Q of discriminant D, an embedding Φ : H ￿→M(2,R) and O(D,N) ⊂ H an Eichler order of level N . Denote by Γ(D,N) the image in PSL(2,R) of Φ(O(D,N)1) ⊂ SL(2,R). As we have seen in Section 1.2.4, Γ(D,N) is a fuchsian group acting on H∗ and the quotient set Γ(D,N)\H∗ has structure of compact Riemann surface. Therefore, it can be seen as the set of points of a projective curve over C. However, this Riemann surface can be naturally identified with the set of complex points of a projective curve over Q, according to the following theorem, which is a particular case of a more general construction due to Shimura (cf. [Shi67], [Shi70]). 1.2 Shimura curves 21 Theorem 1.17. There exists a projective curve X(D,N) defined over Q and a map jD,N : H∗ → X(D,N)(C) with the following properties: (1) The map jD,N gives an analytic isomorphism between Γ(D,N)\H∗ and X(D,N)(C). (2) Let K be a quadratic imaginary field and Λ ⊂ OK an order such that E∗(O(D,N),Λ) ￿= ∅. Let φ ∈ E∗(O(D,N),Λ) and denote by zφ ∈ H the corresponding CM point. Then KΛ = K(jD,N (zφ)), i.e. the coordinates of the point jD,N (zφ) generate the Ring Class Field at- tached to Λ over K. The curve X(D,N) is uniquely determined up to isomorphism by these condi- tions. We call the couple (X(D,N), jD,N ) a canonical model for the Shimura curve. Remark. This correspondence between the set of complex points of the curve and the Riemann surface allows us to translate the notions we had on H to X(D,N). For example we will talk about CM points indistinctly on H or X(D,N)(C). Example. The most simple example of canonical model is the one provided by X0(1) = X(1, 1), the classical modular curve, together with the Klein j-function. The key point in giving Γ(D,N)\H∗ a structure of curve over Q is that its points can be identified with isomorphism classes of principally polarized abelian surfaces with quaternionic multiplication by O(D, 1) and a level N structure. More details can be found in Section 3.5.1, where we are going to explicitly discuss how to recover these polarized abelian surfaces in the case N = 1. The conditions in the theorem are more than enough to characterize the canonical model; if we wished, we could even restrict to the set of CM points by maximal orders. This model has several additional properties which will be useful later on. Proposition 1.9. If D > 1, the curve X(D,N) has no real points. Moreover, we can give a description of the field of rational functions of the curves X(D,N) in terms of functions in H as follows. 22 Quaternion algebras, Shimura curves and automorphic forms Proposition 1.10. Let F ⊂ C be a field. The field {g◦jD,N : g ∈ F (X(D,N))} coincides with the field of meromorphic functions f : H∗ → C ∪ {∞} such that f ◦ γ = f for all γ ∈ Γ(D,N) (i.e. automorphic functions by Γ(D,N) as defined in page 24) and such that f(zφ) ∈ F ·KΛ∪{∞}, for any φ ∈ E∗(O(D,N),Λ). From now on, we will identify the field Q(X(D,N)) with the field of (mero- morphic) automorphic functions with the above condition on the values at CM points. Another property which we will use later on is the Shimura reciprocity law. We will now give an explicit version of it. Given a quadratic imaginary field K and an order Λ ⊂ OK , denote by G = Gal(KΛ|K) = C(Λ). Then, there exists an action of G on the set of equivalence classes of optimal embeddings E∗(O(D,N),Λ)/O(D,N)∗, which we can describe as follows. Fix φ ∈ E∗(O(D,N),Λ) and denote by zφ ∈ H the corresponding fixed point. Let σ ∈ G and consider a fractional ideal a such that ￿ KΛ|K a ￿ = σ. Then, I = φ(a)O(D,N) is an O(D,N)-right ideal and therefore principal, by Theorem 1.5. Let ασ ∈ H∗ (in fact in O(D,N)) be an element of positive norm such that I = ασO(D,N). Theorem 1.18 (Shimura reciprocity law). With the notations above, the ele- ment σ ∈ G maps the optimal embedding i with corresponding fixed point zφ ∈ H to the optimal embedding iσ = α−1σ iασ with fixed point α−1σ z. Equivalently, if f ∈ Q(X(D,N)), then f(zφ)σ = f(α−1σ zφ). In Theorem 1.6 we introduced elements wm ∈ Nor(O(D,N)), for every m￿DN . Consequently, the corresponding elements ωm ∈ PSL(2,R) normal- ize Γ(D,N) and therefore induce analytic involutions on the Riemann surface Γ(D,N)\H∗. Proposition 1.11. The automorphisms induced on the curve X(D,N) by the analytic involutions ωm are defined over Q. The group generated by these invo- lutions is called the Atkin-Lehner group, and will be denoted by WDN . Moreover WDN ∼= (Z/2Z)r, where r is the number of different primes dividing DN . We will both refer to the involution on Γ(D,N)\H∗ and on X(D,N) by ωm. A lot of properties regarding the action of these involutions have been widely studied: their fixed points (cf. [Ogg83]), which we will describe and use in Section 4.1; their relation with the action of the Galois group of the Ring 1.3 Automorphic forms 23 Class Field when acting on a set of CM points by a given order (cf. [Jor81], [GR06]), which we will describe and use in Section 6.2.1, etc. As we observed above, the Klein j-function provides a canonical model for the classical modular curve X0(1). For this function, a classical congruence involving the Atkin-Lehner involution of prime level p holds, namely (j − jp ◦ ωp) · (j ◦ ωp − jp) ≡ 0 (mod p). This congruence generalizes to the well-known Eichler-Shimura congruence formula. Theorem 1.19 (Eichler-Shimura congruence formula, [Shi67]). Let jD be a canonical model for the Shimura curve X(D, 1). There exists a finite set of primes S containing all those primes dividing D such that, if p /∈ S and jpD is the function we obtain from jD when we rise every component to the power p, then (jD − jpD ◦ ωp) · (jD ◦ ωp − jpD) ≡ 0 (mod p). Remark. There are several equivalent ways to interpret the congruence mod p in the previous theorem. The easiest way is to understand it as the reduction of the correspondence given by the image of (jD, jD ◦ ωp) in X(D, 1)×X(D, 1) (cf. [Shi67], [Shi71]). But it can also be understood as a congruence between the values of the functions at certain CM points (cf. [Shi67], [Mor81]). 1.3 Automorphic forms Let Γ be a cocompact fuchsian group and Γ\H∗ the corresponding Riemann surface as introduced in the previous section. Given σ = ￿ a b c d ￿ ∈ GL(2,R) and z ∈ C, we define j(σ, z) = cz + d. For every integer k, every σ ∈ GL+(2,R), and every function f : H → C, we define (f |kσ)(z) = det(σ)k/2j(σ, z)−kf(σ(z)). Then, for another τ ∈ GL+(2,R), we have that f |kστ = (f |kσ)|kτ. 24 Quaternion algebras, Shimura curves and automorphic forms Definition. Let k be an even integer. A C-valued function f on H is called an (meromorphic) automorphic form of weight k with respect to Γ if it satisfies the following three conditions: (a) f is meromorphic on H; (b) f |kγ = f for all γ ∈ Γ; (c) f is meromorphic at every cusp of Γ. Remark. Note that since k is even, f |kγ does not depend on the representative in SL(2,R) we choose. The last condition only applies when Γ has cusps. For example, when we deal with the groups Γ(D,N) introduced above, this translates into D = 1. The condition of being meromorphic at every cusp can be translated as follows. For a given cusp s, we choose a transformation ρ sending s to ∞. Then, the function f |kρ−1 is invariant by a translation of length h. Therefore, there exists a suitable neighborhood of 0 such that f |kρ−1(z) = Φ(e2πiz/h). We say that f is meromorphic at s if Φ is meromorphic at 0. In particular, if we consider a power expansion of Φ around 0, we obtain a Fourier expansion for f |kρ−1, f |kρ−1(z) = ￿ n≥n0 cne 2πinz/h. If k = 0, we recover the notion of automorphic function with respect to Γ. Definition. Let us define Ak(Γ) as the set of all (meromorphic) automorphic forms of weight k with respect to Γ, in particular A0(Γ) is the set of automorphic functions. We denote by Gk(Γ) the set of all holomorphic (including the cusps) automorphic forms of weight k with respect to Γ and by Sk(Γ) the subset of Gk(Γ) consisting of those forms vanishing at the cusps. The elements of Sk(Γ) are usually called cusp forms. Note that if Γ has no cusps, which is the case we are more interested in, Gk(Γ) = Sk(Γ). In the case of the groups Γ(1, N), instead of referring to these functions as automorphic functions and automorphic forms, we call them modular functions and modular forms. Observe that A0(Γ) is the field of automorphic functions and can be iden- tified with the field of meromorphic functions on the Riemann surface Γ\H∗. 1.3 Automorphic forms 25 Therefore it is a field of transcendence degree 1 over C. Moreover all the spaces Ak(Γ) are vector spaces of dimension one over A0(Γ). On the other hand, Gk(Γ), Sk(Γ) are C-vector spaces of finite dimension. A formula for the dimension of these spaces is given in the following theorem. Theorem 1.20. Let g be the genus of Γ\H∗, e∞ the number of inequivalent cusps and e1, . . . , er the orders of the elliptic cycles of Γ. Then, for an even integer k, dimCGk(Γ) =  (k − 1)(g − 1) + k2 e∞ + r￿ i=1 ￿k(ei−1)2ei ￿, if k > 2, g + e∞ − 1, if k = 2, e∞ > 0, g, if k = 2, e∞ = 0, 1, if k = 0, 0, if k < 0, and dimC Sk(Γ) =  (k − 1)(g − 1) + (k2 − 1)e∞ + r￿ i=1 ￿k(ei−1)2ei ￿, if k > 2, g, if k = 2, 1, if k = 0, e∞ = 0, 0, if k = 0, e∞ > 0, 0, if k < 0. From now on we will restrict ourselves to the case of the groups Γ(D,N). Hence, we fix an indefinite quaternion algebra of discriminant D and an Eichler order O = O(D,N) of level N , together with an embedding Φ : H →M(2,R) and the corresponding group Γ(D,N) ⊂ PSL(2,R). For the field of automorphic functions, A0(Γ(D,N)), we have already seen that there exists a canonical subfield A0(Γ(D,N))Q = Q(X(D,N)) such that A0(Γ(D,N)) = A0(Γ(D,N))Q ⊗Q C. Now we are going to see how the same is done with Ak(Γ(D,N)). In the classical modular case, this is done using the coefficients of the Fourier expansions introduced above. However, in the general case, Fourier expansions are no longer available. Notice that Ak(Γ(D,N)) is a vector space of dimension 1 over A0(Γ(D,N)) and the same has to happen for the corresponding subspaces over Q. Therefore, 26 Quaternion algebras, Shimura curves and automorphic forms we only need to find a suitable element for each of these spaces. Moreover, if f ∈ A2(Γ(D,N))Q, then fk ∈ A2k(Γ(D,N))Q, therefore it will be sufficient to choose an element in A2(Γ(D,N))Q. In order to do so, we only need to note that, if f ∈ A0(Γ(D,N)), then dfdz ∈ A2(Γ(D,N)). Therefore we could take dfdzA0(Γ(D,N))Q for any non-constant function f ∈ A0(Γ(D,N))Q. The fact that A0(Γ(D,N))Q is an extension of Q of transcendence degree 1 and such that Qab ∩ A0(Γ(D,N))Q = Q ensures the independence of the chosen f . However, in order for the definition to agree with the modular classical structure we will take A2(Γ(D,N))Q = 12πi df dzA0(Γ(D,N))Q for any non-constant function f ∈ A0(Γ(D,N))Q, for which all previous assertions also hold. From this, we obtain sets Gk(Γ(D,N))Q and Sk(Γ(D,N))Q intersecting with the set of holomorphic forms or holomorphic forms vanishing at the cusps. Replacing Q with a number field K we obtain the definition of automorphic forms over K. From the definition it is clear that Ak(Γ(D,N)) = Ak(Γ(D,N))Q ⊗Q C. However, although it is not obvious in any way, even in the case of modular forms, we can choose a basis of any of the spaces Gk(Γ(D,N)) and Sk(Γ(D,N)) consisting of elements of the corresponding spaces over Q. Finally we are going to introduce a family of Hecke operators acting on the corresponding spaces of automorphic forms. Given a positive integerm such that gcd(m,DN) = 1, we considerO(D,N)￿m the set of primitive elements of normm and we choose a set αi ∈ GL+(2,R), i = 1, . . . , r, corresponding by the embed- ding Φ to a system of representatives of the quotient set O(D,N)1\O(D,N)￿m. Then, we can define an operator Tm : Ak(Γ(D,N))→ Ak(Γ(D,N)) such that Tmf = det(α) k/2−1 r￿ i=1 f |kαi. This operator acts as well on the corresponding sets of holomorphic automor- phic forms and of cusp forms. It can be seen that there exists a basis for Sk(Γ(D,N))Q consisting of eigenvectors for all the Tm, usually called eigen- forms. We will not go into detail in the properties of the Hecke operators since just the definition will suffice for our purposes. More details on how to explicitly compute their action on the spaces of cusps forms will be given in Chapter 6. Chapter 2 Kroneckerian polynomials Let H be the upper half-plane and j : H→ C the classical modular j-function, which is a generator of the field of rational functions of the modular curve X0(1) over Q. For any integer N ≥ 1, there exists an irreducible polynomial ΦN (X,Y ) ∈ Z[X,Y ] such that ΦN (j(τ), j(Nτ)) = 0 for all τ ∈ H. For N = 1, Φ1(X,Y ) = X −Y , and for N > 1, ΦN (X,Y ) is a symmetric polynomial which provides a singular model for the modular curveX0(N) in P1Q×P1Q. The equation ΦN (X,Y ) = 0 was already studied by Kronecker and is classically called the modular equation of level N (cf. [Lan87]). Let HD be a rational indefinite quaternion algebra of discriminant D > 1, and fix an embedding Φ : HD ￿→ M(2,R). For a positive integer M coprime to D, let X(D,M) be the Shimura curve attached to an Eichler order of level M in HD, O = O(D,M), and assume that the quotient X of X(D,M) by a certain subgroup W of the Atkin-Lehner group of level DM has genus 0. X is a curve defined over Q attached to a certain group Γ, but in contrast to the classical modular case we introduced above, this curve may not have rational or even real points. Therefore, there exists in general no generator of the field of rational functions defined over Q. Moreover, the classical description of modular functions by means of Fourier expansions cannot be extended to this case due to the absence of cusps. Let us take X as our base curve. For a positive integer N coprime to DM and a complex generator f of the field of rational functions, which we will call a Hauptmodul, we construct a polynomial ΨΓ,N,f (X,Y ) ∈ C[X,Y ] playing the role of the modular polynomial of level N , which we call the kroneckerian polynomial of level N . It provides a complex 27 28 Kroneckerian polynomials model for the corresponding quotient of the Shimura curve of discriminant D and level MN , X(D,MN)W . Note that this curve has in general genus greater than 0. In Theorem 2.1 we prove that this polynomial is irreducible, symmetric (if N > 1) and that, if the function f is defined over a number field K, then its coefficients can also be taken in K. In Proposition 2.5 we study the roots of the polynomial ΨΓ,N,f (X,X) together with their multiplicities and we obtain a formula for its decomposition similar to the factorization in class equations which holds in the modular case, providing therefore a way to compute the exact values of the function f at certain CM points. Then, for the particular curves X(D, 1), D = 6, 10, 22, we introduce some specific Hauptmoduln tD defined over some explicit quartic number fields. In the case D = 6, this Hauptmodul is the one introduced in [BT07b]. For these choices, we prove that the model provided by ΨΓ,N,tD descends to Q, even though these Hauptmoduln are not defined over Q, and that a congruence sim- ilar to the Kronecker congruence formula holds. Our approach is similar to the usual construction for the modular case, cf. [Cox89, Lan87], except for the proof of the congruence relation, which in our case relies on the Shimura reciprocity law. In particular, the explicit de- scription of the polynomials ΨΓ,N,f will allow us to prove additional arithmetic properties and to effectively compute them without making any assumption on the genus of X(D,MN)W , using only a differential equation satisfied by the Hauptmodul, which provides expansions for it around suitable CM points. The explicit computation of these polynomials, together with its application to the determination of the values of f at certain CM points, will be considered in Chapter 6. 2.1 The polynomials ΨΓ,N,f Let HD be the rational indefinite quaternion algebra of discriminant D > 1 and Φ : HD ￿→M(2,R) an embedding of HD into the real algebra of 2× 2 matrices. Moreover, we also fix a positive integer M coprime to D, a maximal order O(D, 1) inside HD and an Eichler order of level M , O = O(D,M) ⊆ O(D, 1), with Z-basis {1, I, J,K}. We will be mainly interested in the case M = 1; however, at some points it will be useful to have considered this more general setting, which only amounts to slightly more involved notations and statements. For every positive integer N coprime to DM , we consider Eichler orders of level MN contained in O(D,M). Denote O = O(1) and O(N) = O(D,MN); 2.1 The polynomials ΨΓ,N,f 29 then, for every integer k, we write O(N)k for the set of quaternions of norm k in O(N). In particular, O(N)1 coincides with the group of units of norm 1 of O(N). This group will be identified with its image in SL(2,R) under Φ. We denote Γ(N) = Γ(D,MN), the image of Φ(O(N)1) by the standard projection SL(2,R) → PSL(2,R). If k = l is a prime, the notation Ol can both denote the elements of norm l or O ⊗ Zl. When confusion may arise, we will denote O(l) = O ⊗ Zl. Consider in H the usual action of the group PSL(2,R) via homographic transformations. Then, as we have seen in Section 1.2.7, the corresponding action of the group Γ(N) gives rise to a compact Riemann surface, Γ(N)\H, which we can identify canonically with the set of complex points of a complete non-singular curve defined over Q, which we denote by X(N) = X(D,MN), the Shimura curve of discriminant D and level MN . Let m ≥ 1 be an integer such that m divides exactly DMN , namely, m divides DMN and it is coprime to DMN/m. In this situation, we choose an element wm ∈ O(N)m which normalizes O(N). This element is not unique but, for a fixed m, all these wm define the same automorphism of Γ(N)\H. This au- tomorphism is involutive (the identity when m = 1). We will denote by ωm the corresponding element of PSL(2,R) as well as the corresponding automorphism of Γ(N)\H. It follows from the description given before Theorem 1.6, that the element wN allows us to write the order O(N) as O(N) = O ∩ w−1N OwN and, therefore, its group of units of norm 1 as O(N)1 = O1 ∩ w−1N O1wN . Recall that, by Proposition 1.11, the involutions ωm such that m divides exactly DMN form the Atkin-Lehner group WDMN . The structure of this group is given by WDMN ∼= Cr2 , where C2 is the cyclic group of order 2 and r is the number of distinct primes dividing DMN . These involutions are, in fact, rational automorphisms. For any subgroupW ⊆WDMN , we denote by X(N)W the corresponding quotient curve by W . In the situation above, the particular involution ωN will play a central role. To define the kroneckerian polynomials of level N we work with the groups Γ(D,M)￿W ￿, the groups generated inside PSL(2,R) by Γ(D,M) and a sub- group W of the full Atkin-Lehner group WDM of level DM . For a subgroup W ⊆WDM we will abbreviate Γ = Γ(D,M)￿W ￿ when there is no risk of confu- sion. For every positive integer N coprime to DM , let O￿N ⊆ ON ⊆ O be the subset of primitive elements of norm N , O￿N = {a+ bI + cJ + dK ∈ ON : gcd(a, b, c, d) = 1} ⊆ ON . 30 Kroneckerian polynomials Note that this definition does not depend on the Z-basis {1, I, J,K} which we consider in O and that, if N is squarefree, then O￿N = ON . Now, as before, let wN ∈ ON be an element that represents the involution of level N on X(N) and consider the sets of orbits by the action of the groups O(N)1 and O1 R(N) = O(N)1\O1 and R˜(N) = O1\O1wNO1. By abuse of notation, we also write R(N) and R˜(N) for a set of representatives of each of these sets. The first result we prove is the following. Proposition 2.1. Let N be a positive integer coprime to DM . Then, (1) There exists a bijection between R(N) and R˜(N). (2) The cardinality of the sets R(N) and R˜(N) is given by the classical ψ func- tion: #R(N) = #R˜(N) = ψ(N) = N ￿ p|N p prime (1 + p−1). (3) O1wNO1 = O￿N . Proof. (1) Consider the map r : O1 −→ O1\ON defined by r(γ) = O1wNγ, for any γ ∈ O1. Then, r factors through the quotient set R(N) and the corre- sponding map r : R(N) −→ O1\ON is injective: for γ, γ￿ ∈ O1, the equality O1wNγ = O1wNγ￿ holds if and only if γ￿γ−1 ∈ O1 ∩ w−1N O1wN = O(N)1, and this is equivalent to O(N)1γ￿ = O(N)1γ. Moreover, the image of r is, clearly, R˜(N), which yields the first statement. (2) The cardinality of R(N), and therefore, that of R˜(N), is computed in [Vig80] and [AB04]. (3) From the local description of the ideal wNO(N) given in page 9, it follows that wN ∈ O￿N ; then, we have that O1wNO1 ⊆ O￿N and this fact gives us an inclusion R˜(N) ￿→ O1\O￿N . (∗) Now we will see that R˜(N) = O1\O￿N by showing that the cardinality of the two sets coincides. To compute the cardinality of the right hand side of (∗), we will use again the local description of the orders and the involutions wN . If we 2.1 The polynomials ΨΓ,N,f 31 denote by (O ⊗ Zl)N the set of elements in O ⊗ Zl which have norm N up to units of Zl, then the natural map ON −→ ￿ l prime (O ⊗ Zl)∗\(O ⊗ Zl)N factors through O1\ON giving rise to an inclusion. Denote by (O ⊗ Zl)￿N the set of primitive elements of (O ⊗ Zl)N ; then, the restriction to O￿N gives an injection O1\O￿N −→ ￿ l prime (O ⊗ Zl)∗\(O ⊗ Zl)￿N . To finish the proof, it suffices to prove that the cardinality of the right hand side is ψ(N). If l ￿ N , we have that (O ⊗ Zl)￿N = (O ⊗ Zl)∗ and hence the quotient is trivial. Otherwise, if k ≥ 1 is the exact power of l that divides N , then the set (O ⊗ Zl)￿N = (O ⊗ Zl)￿lk can be identified with the set of elements{X ∈ M(2,Zl)￿ : detX ∈ lkZ∗l } when (O ⊗ Zl)∗ is identified with M(2,Zl)∗. Proceeding in the same way as it is computed in the modular case (cf. [Lan87]), a set of representatives for this quotient is￿￿ lr b 0 lk−r ￿ ∈M(2,Z) : 0 ≤ r ≤ k, 0 ≤ b < lk−r, gcd(lk, lk−r, b) = 1 ￿ , which has cardinality ψ(lk). Therefore, the set￿ l prime (O ⊗ Zl)∗\(O ⊗ Zl)￿N has cardinality ψ(N), as we wanted to see. Now, let G ⊂ PSL(2,R) be a subgroup and consider the set of orbits R(G,N) = (G ∩ ω−1N GωN )\G. As before, it will denote both the set itself and a set of representatives. For a subgroup W ⊆WDM , write Γ = Γ(D,M)￿W ￿. Corollary 2.1. (1) The projection O1 −→ Γ(D,M) induces a bijection R(N) −→ R(Γ(D,M), N) which is compatible with the right action of O1 and Γ(D,M), respectively. 32 Kroneckerian polynomials (2) The inclusion Γ(D,M) ￿→ Γ induces a bijection R(Γ(D,M), N) −→ R(Γ, N) which is compatible with the right action of Γ(D,M) on both sets. Proof. (1) The proof of this statement is obtained by applying the projection SL(2,R) −→ PSL(2,R). (2) Consider the composition i : Γ(D,M) ￿→ Γ −→ R(Γ, N). It follows from Theorem 1.6 that any element α ∈ Γ can be written as α = γω for γ ∈ Γ(D,M), ω ∈ W in a unique manner. Moreover, if we choose W as a subgroup inside WDMN , then ωN commutes with the elements of W up to an element in Γ(D,M) ∩ ω−1N Γ(D,M)ωN . Therefore, we have that Γ ∩ ω−1N ΓωN = (Γ(D,M) ∩ ω−1N Γ(D,M)ωN )￿W ￿ and i is easily seen to be surjective. Moreover, i factors through the quotient set R(Γ(D,M), N) giving rise to a bijection which preserves the right action of Γ(D,M), as desired. We will need as well some results that show us how to write elements of a given prime power norm as products of elements of smaller norms in a somehow unique way. First, we will prove the following lemma. Lemma 2.1. Let k ≤ r be positive integers, p a prime not dividing DM and O(pk) ⊃ O(pr) Eichler orders of levels Mpk and Mpr, respectively, contained in a fixed Eichler order O of level M . Then, for any choice of representa- tives of the corresponding Atkin-Lehner involutions, wpk and wpr , we have that wpkwpr , wprwpk ∈ pkO￿pr−k . In particular, wpr ∈ O￿pr−kwpk ∩ wpkO￿pr−k . Proof. By Proposition 1.3 and Theorem 1.4, we can consider an isomorphism ϕp : O ⊗ Zp ∼= M(2,Zp) such that ϕp(O(pk) ⊗ Zp) = ￿ Zp Zp pkZp Zp ￿ and ϕp(O(pr)⊗ Zp) = ￿ Zp Zp prZp Zp ￿ . Then, ϕp(wpk) ∈ ￿ Zp Zp pkZp Zp ￿￿ 0 1 pk 0 ￿ = ￿ 0 1 pk 0 ￿￿ Zp Zp pkZp Zp ￿ and ϕp(wpr ) ∈ ￿ Zp Zp prZp Zp ￿￿ 0 1 pr 0 ￿ = ￿ 0 1 pr 0 ￿￿ Zp Zp prZp Zp ￿ . 2.1 The polynomials ΨΓ,N,f 33 Therefore, ϕp(wpkwpr ) ∈ pkϕp(O ⊗ Zp)￿ and we obtain the first claim. As a consequence, if we replace the representative wpk by wpk , we obtain wprwpk ∈ pkO￿pr−k , and therefore wpr ∈ O￿pr−kwpk . Proposition 2.2. For an Eichler order O of level M , a positive integer k and a prime p not dividing DM , consider Eichler orders of levels Mpk and Mp, O(pk) ⊂ O(p) ⊂ O, and an element wpk ∈ O(pk)￿pk giving rise to the Atkin- Lehner involution of level pk. Denote by Γ˜(pk) ⊂ Γ˜(p) ⊂ Γ˜ the corresponding groups of units of norm 1 and Γ˜￿ = Γ˜￿ Γ˜(p). Fix as well a set of representatives R(Γ˜, pk) of Γ˜(pk)\Γ˜ and R￿(Γ˜, pk) a set of representatives of Γ˜(pk)\Γ˜￿. Now, for a positive integer n ≥ 2, consider the map ￿ : Γ˜× n+1· · · ×Γ˜ → Opnk (γ0, . . . , γn) ￿→ γ0wpkγ1wpk · · ·wpkγn−1wpkγn. Then, (1) ￿ is a surjective map. (2) ￿ induces a bijection between Γ˜\O￿pkn and {1}×R￿(Γ˜, pk)n−1 ×R(Γ˜, pk). (3) O￿pkn = ￿(Γ˜× (Γ˜￿)n−1 × Γ˜) = ￿(Γ˜×R￿(Γ˜, pk)n−1 ×R(Γ˜, pk)). (4) For i ≤ k, an element α = ￿(γ0, . . . , γn) ∈ O￿pkn can be written as α = piα0 for some integral element α0 if and only if γnγ0 ∈ Γ˜(pi). Proof. (1) By Proposition 2.1, we have that O￿pk = Γ˜wpk Γ˜, or, equivalently, that O￿pk = Γ˜αΓ˜, for any α ∈ O￿pk . Consequently, it is only necessary to prove that for any 0 ≤ i ≤ nk, there exists a primitive element of norm pi in the image of ￿, which follows from the previous lemma. (2) The inclusion O￿pkn ⊂ ￿(Γ˜ × (Γ˜￿)n−1 × Γ˜) follows from (1) together with the lemma above. Recall as well that the quotient set Γ˜\O￿pkn has ψ(pkn) elements and this coincides with the cardinality of {1}×R￿(Γ˜, pk)n−1×R(Γ˜, pk). Now, we claim that ￿ induces a surjection from {1}×R￿(Γ˜, pk)n−1×R(Γ˜, pk) to Γ˜\O￿pkn as follows. We have seen that any element in O￿pkn can be written as 34 Kroneckerian polynomials γ0wpkγ1wpk · · · γn−1wpkγn with γ1, . . . , γn−1 ∈ Γ˜￿. Then, since wpk normalizes Γ˜(pk), we can replace γn by an element in R(Γ˜, pk), which changes γn−1 ∈ Γ˜￿ by γ￿n−1 ∈ Γ˜￿. Now, in the same manner, we can replace γ￿n−1 by an element in R￿(Γ˜, pk). Iterating, we obtain the claim. Finally, since both sets are finite with the same cardinal we can conclude that the map is indeed a bijection. (3) It follows immediately from (2), where we have already proved that O￿pkn ⊂ ￿(Γ˜ × (Γ˜￿)n−1 × Γ˜) = ￿(Γ˜ × R￿(Γ˜, pk)n−1 × R(Γ˜, pk)) and also that ￿(Γ˜×R￿(Γ˜, pk)n−1 ×R(Γ˜, pk)) ⊂ O￿pkn . (4) First of all, observe that, if α = ￿(γ0, . . . , γn) with γnγ0 ∈ Γ˜(pi), then α = (wpiγn) −1wpiγnα = (wpiγn)−1wpiγnγ0wpkγ1 · · · γn−1wpkw−1pi (wpiγn) and since γnγ0 ∈ Γ˜(pi), wpiγnγ0wpk ∈ piOpk−i and wpkw−1pi ∈ Opk−i . Thus we have that α ∈ pi(wpiγn)−1Opn−2(wpiγn) and the result follows. To obtain the other implication, fix γ0, . . . , γn such that α = ￿(γ0, . . . , γn). Now, we are going to prove that the set {γ ∈ Γ˜ : ￿(γ, γ1, . . . , γn−1, γn) = piβ, for a certain integer β} constitutes a unique class in Γ˜/Γ˜(pi). Since both γ0 and γ−1n satisfy this prop- erty, we will be done. By Proposition 1.3 and Theorem 1.4, completing at p, we can identify O⊗Zp withM(2,Zp) and (O(pi)⊗Zp)1 with Γ0(pi)p. This gives a bijection between (O⊗Zp)1\(O⊗Zp)￿pkn and SL(2,Zp)\M(2,Zp)￿pkn and we can identify wpkn with ￿ 0 1 pkn 0 ￿ . An element of the form ￿(γ, γ1, . . . , γn−1, γn) is pi times an integer, if and only if the same holds in the completion, and therefore we can work in this new setting. The usual set of representatives of SL(2,Zp)\M(2,Zp)￿pkn is given by the matrices￿￿ ps x 0 pkn−s ￿ : 0 ≤ s ≤ kn, 0 ≤ x < pkn−s, gcd(ps, pkn−s, x) = 1 ￿ , but it will be better for our purposes to consider￿ As,x = ￿ ps x 0 pkn−s ￿ : 0 ≤ s ≤ (n− 1)k, 0 ≤ x < pkn−s, gcd(ps, x) = 1 ￿ ∪ ￿ As,x = ￿ yps −1 pkn 0 ￿ : (n− 1)k < s ≤ kn, 0 ≤ x, y < pkn−s, gcd(pkn−s, x) = 1, xy + 1 ≡ 0 (mod pkn−s) ￿ . 2.1 The polynomials ΨΓ,N,f 35 This last system of representatives has the additional property that it cor- responds to elements in the completion of ￿({1}× (Γ˜￿)n−1 × Γ˜). Therefore, all we have to prove is that for each one of these representatives, As,x, the set of γ ∈ SL(2,Zp) such that pi| tr(γ · As,x) is a class in SL(2,Zp)/Γ0(pi)p. Write γ = ￿ a b c d ￿ ∈ SL(2,Zp) and observe that pi| tr(γ ·As,x) if and only if ￿ γ ∈ Γ0(pi)p, if i ≤ s, c = psc0, i.e. γ ∈ Γ0(ps)p, and pi−s|a+ c0x, if i > s. Since for γ￿ = ￿ a￿ b￿ c￿ d￿ ￿ ∈ SL(2,Zp) such that c￿ = psc￿0 and pi−s|a￿ + c￿0x, we have γ￿−1γ ∈ Γ0(pi)p, we obtain the result. Proposition 2.3. Let D and M be integers as before and fix O an Eichler order of level M . Let Λ be an order in a quadratic imaginary field and φ : Λ→ O an optimal embedding. Let s be an integer dividing exactly DM , that is s|DM and gcd(s,DM/s) = 1. Then, if λ ∈ Λ has norm sN and ws · φ(λ) ∈ sO, we have that λ2 ∈ sΛ. Reciprocally, if s is squarefree and λ is such that λ2 ∈ sΛ, then ws · φ(λ) ∈ sO. Proof. The first assertion is clear: if ws · φ(λ) ∈ sO, it follows that φ(λ) ∈ wsO and therefore that φ(λ)2 = φ(λ2) ∈ sO, since ws normalizes O. Thus, since φ is optimal, λ2 ∈ sΛ. On the other hand, if we assume that s is squarefree, we need to show that we can go the other way around. We can work locally for every l|s. If l|D, then it is clear that φ(λ)l ∈ O(l) ￿ O∗(l) = ωlO(l). Otherwise, if l|M , then the element φ(λ)l = γ0wlγ1 for certain γ0, γ1 ∈ O(D, 1)∗(l). In particular, φ(λ)l ∈ O(D, l)(l) = O(D,M)(l) if and only if wlφ(λ)lwl ∈ lO(D,M)(l), which by the proposition above holds if and only if γ0 or γ1 ∈ O(D,M)∗(l). However, since λ2 ∈ sΛ, we obtain that γ0γ1 ∈ O(D,M)∗(l). Therefore φ(λ)l ∈ ωlO(l) and the result follows. Remark. In the non-squarefree case, if we fix M ￿ such that M/M ￿2 is square- free, the elements λ such that λ2 ∈ sΛ are of the form γωsOγ with γ ∈ Γ(M/M ￿). Since we consider the embeddings up to conjugation by Γ(M), the condition ωs · φ(λ) ∈ sO, fixes a conjugation class among Γ(M)\Γ(M/M ￿). 36 Kroneckerian polynomials From now on, fix W ⊆ WDM , consider the group Γ = Γ(D,M)￿W ￿ and as- sume that the curve X(D,M)W has genus 0. Thus, there exists a Hauptmodul, i.e. a Γ-automorphic function f : H → C ∪ {∞} such that the field of rational functions of the complex curve X(D,M)WC is C(f). Definition. Let f be a Hauptmodul for X(D,M)W . For each positive integer N coprime to DM , we define Ψ˜Γ,N,f (X, f) = ￿ γ∈R(Γ,N) (X − f ◦ ωN ◦ γ), where ωN is the transformation attached to an element in ON representing the Atkin-Lehner involution of level N on X(D,MN). The kroneckerian polynomial of level N for f will be a modification of it, but first we need to study some of the properties of this one, which we collect in the following proposition. Proposition 2.4. The polynomial Ψ˜Γ,N,f (X, f) belongs to C(f)[X] and it is irreducible, monic and has degree ψ(N). Proof. The left action of Γ on H gives rise to a right action on the field of meromorphic functions H −→ P1(C). Being f a Γ-automorphic function, this action is trivial on the field C(f). On the other hand, by using Corollary 2.1 and Proposition 2.1, we see that the action can be restricted to the set Ω = {f ◦ ωN ◦ γ : γ ∈ R(Γ, N)} and that it is transitive on it. Therefore, Γ acts on the right as a group of automorphisms on the field C(f,Ω). Now, the coefficients of Ψ˜Γ,N,f (X, f), as a polynomial in X, are the ele- mentary symmetric polynomials of Ω; thus, we obtain that these symmetric polynomials are invariant under transformations z ￿→ δz, for δ ∈ Γ, and there- fore automorphic with respect to this group. Since any Γ-automorphic function is a rational function of f , we get that Ψ˜Γ,N,f (X, f) ∈ C(f)[X]. Moreover, by the transitivity of the action on the set Ω, one has that Ψ˜Γ,N,f (X, f) is the minimal polynomial of f ◦ ωN over the fixed field by Γ, C(f,Ω)Γ, and, since C(f) ⊆ C(f,Ω)Γ, it follows that Ψ˜Γ,N,f (X, f) = Irr(f ◦ ωN ,C(f)). Thus, it is irreducible. 2.1 The polynomials ΨΓ,N,f 37 Now, Ψ˜Γ,N,f (X, f) is a monic polynomial in C(f)[X] and therefore we can multiply it by a certain polynomial in C[f ] to remove the denominators of the coefficients without adding any common factors; in this way, we obtain an irre- ducible polynomial ΨΓ,N,f (X,Y ) ∈ C[X,Y ]. Definition. For a subgroup W ⊆ WDM , we take the group Γ = Γ(D,M)￿W ￿ and a Hauptmodul f for Γ. Then, for a positive integer N coprime to DM , we call a polynomial ΨΓ,N,f (X,Y ) ∈ C[X,Y ] as above a kroneckerian polynomial of level N attached to the Hauptmodul f for the group Γ. Note that, for every positive integer N coprime to DM , the polynomial ΨΓ,N,f (X,Y ) is unique up to multiplication by a non-zero complex constant. Therefore, all statements involving kroneckerian polynomials should be under- stood up to multiplication by such a constant, even if it is not made completely explicit. We state now the basic properties of the kroneckerian polynomials. Theorem 2.1. Let f be a Hauptmodul for X(D,M)WC . For every positive inte- ger N coprime to DM , the kroneckerian polynomial attached to f , ΨΓ,N,f (X,Y ), is an irreducible polynomial of degree ψ(N) in each indeterminate. If N = 1, we can take ΨΓ,1,f (X,Y ) = X − Y ; if N > 1, then ΨΓ,N,f (X,Y ) is symmetric. Moreover, if f is a Hauptmodul defined over a number field K, then, after a suitable choice of the constant of its definition, ΨΓ,N,f (X,Y ) ∈ K[X,Y ]. Proof. The irreducibility has already been established. Moreover, it is clear that, if N = 1, then we can take ΨΓ,1,f (X,Y ) = X − Y . So, we assume from now on thatN > 1. To prove that ΨΓ,N,f (X,Y ) has degree ψ(N) in each indeterminate, it suffices to see that it is symmetric, because, by construction, it has degree ψ(N) as a polynomial in X. Now, to prove the symmetry, we will use that ωN is an involution. ΨΓ,N,f (X, f) is an irreducible polynomial over C(f) which has f ◦ ωN as a root. Thus, we have the identity ΨΓ,N,f (f(ωNz), f(z)) = 0, for each z ∈ H. Applying this to ωNz ∈ H, we have ΨΓ,N,f (f(ω2Nz), f(ωNz)) = 0; therefore, ΨΓ,N,f (f(z), f(ωNz)) = 0, for all z ∈ H. That is, f ◦ ωN is a root of the polynomial ΨΓ,N,f (f,X), which is the polynomial we started with, but with the variables exchanged. By the irreducibility of ΨΓ,N,f (X, f), we obtain that it divides ΨΓ,N,f (f,X) in C(f)[X]. Thus, there exists g(X,Y ) ∈ C(Y )[X] such that ΨΓ,N,f (f,X) = g(X, f)ΨΓ,N,f (X, f); using Gauss’ Lemma, we see that g(X,Y ) ∈ C[Y,X]. Then, we have that ΨΓ,N,f (f,X) = g(X, f)g(f,X)ΨΓ,N,f (f,X), 38 Kroneckerian polynomials which implies that g(X,Y ) = ±1. If it were −1, then ΨΓ,N,f (f,X) = −ΨΓ,N,f (X, f), which would yield that ΨΓ,N,f (f, f) = 0 and, by the irreducibility, the degree in X would be 1 and this is impossible since it has degree ψ(N) > 1. Therefore, g(X, f) = 1 and we get the symmetry. Finally, assume that f is a Hauptmodul defined over a number field K ⊂ C, in the sense that the field of K-rational functions of the curve X(D,M)W is K(f). Since the involutions ωN are automorphisms of X(D,MN)W defined over Q, both f and f ◦ ωN are defined over K; in other words, we have an extension of fields K(f, f ◦ωN ) | K(f). Since the minimal polynomial of f ◦ωN for this extension is also Ψ˜Γ,N,f (X, f), it is now obvious that ΨΓ,N,f (X, f) can be taken such that ΨΓ,N,f (X, f) ∈ K[X, f ]. Corollary 2.2. Let K be a number field and let f be a Hauptmodul for the curve X(D,M)W defined over K. Then, for every positive integer N coprime to DM , the fields of K-rational functions of X(N)W and X(N)￿W,ωN ￿ are the fields K(X(N)W ) = K(f, f ◦ ωN ) and K(X(N)￿W,ωN ￿) = K(f · f ◦ ωN , f + f ◦ ωN ). Proof. Since K(f) = K(X(D,M)W ) and K(f, f ◦ ωN ) ⊆ K(X(N)W ), the first result follows from the equality of degrees [K(f, f ◦ ωN ) : K(f)] = [K(X(N)W ) : K(f)] = ψ(N). The other equality of fields follows from the fact that we can identify the function field of K(X(N)￿W,ωN ￿) as the subfield of K(X(N)W ) fixed by ωN , which acts interchanging f and f ◦ ωN . Corollary 2.3. Let K be a number field and let f be a Hauptmodul defined over K for X(D,M)W . For every integer N > 1 coprime to DM , the polynomial ΨΓ,N,f (X + √ Y ,X −√Y ) belongs to K[X,Y ], is irreducible over C and provides a model over K for the curve X(N)￿W,ωN ￿. Proof. The expression p(X, √ Y ) = ΨΓ,N,f (X + √ Y ,X − √Y ) clearly belongs to K[X, √ Y ]. Moreover, because of the symmetry of ΨΓ,N,f (X,Y ), we obtain 2.1 The polynomials ΨΓ,N,f 39 that p(X, √ Y ) ∈ K(X,√Y ) is fixed by the Galois group of K(X,√Y )|K(X,Y ). Hence, it lies in K(X,Y ) ∩K[X,√Y ] = K[X,Y ]. Now we will prove the irre- ducibility in C[X, √ Y ], which implies the same result in C[X,Y ]. Consider the isomorphism of algebras φ : C[X, √ Y ]→ C[T, U ] such that φ(X) = (T + U)/2, φ( √ Y ) = (T − U)/2 and then φ(p(X, √ Y )) = p((T + U)/2, (T − U)/2) = ΨΓ,N,f (T, U), for which we already have proved the irreducibility. Finally, in the previous corollary we have seen that K(X(D,MN)￿W,ωN ￿) = K(f · f ◦ ωN , f + f ◦ ωN ) = K((f + f ◦ ωN )/2, ((f − f ◦ ωN )/2)2) and p( f+f◦ωN2 , ￿ ((f − f ◦ ωN )/2)2) = 0, from which we obtain the last asser- tion. 2.1.1 Relation to class equations As in the modular case, the roots of the polynomial ΨΓ,N,f (X,X) are related to the values of f at certain CM points. In particular, this provides a method to compute the values of f at an explicit set of CM points, which we will exploit in Chapter 6. Some of the most basic properties of the q-expansions introduced in Section 3.3 will be used in the proof. Proposition 2.5. Let W be a subgroup of WDM , Γ = Γ(D,M)￿W ￿ and f a Hauptmodul for the curve X(D,M)WC . Consider the set S = {s : ωs ∈W}. Given τ0 a CM point by an order Λ in a quadratic field K, the multiplicity of f(τ0) as a root of ΨΓ,N,f (X,X) is given by r(Λ, N, S) = #{x ∈ Λ￿ : ωsφ(x) ∈ sO(D,M)N for a certain s ∈ S}/ ∼ = #{x ∈ Λ￿ : ωsφ(x) ∈ sO(D,M)N for a certain s ∈ S}/2e(τ0), where x ∼ y if and only if xy ∈ NΛ, e(τ0) denotes the elliptic order of τ0 and · denotes the conjugation. In particular, z0 ∈ C is a root of the polynomial ΨΓ,N,f (X,X) ∈ C[X] if and only if z0 = f(τ0) for a complex multiplication point τ0 ∈ H attached to an optimal embedding φ : Λ ￿→ O(D,M) such that there exists a primitive element x ∈ Λ of norm sN for a certain s ∈ S for which ωsφ(x) ∈ sO(D,M) (cf. Proposition 2.3). 40 Kroneckerian polynomials Proof. First of all, assume that f ◦ ωN ◦ γ(τ0) ∈ C for all γ ∈ R(Γ, N). Then, since ΨΓ,N,f (X, f) = P (f) ￿ γ(X − f ◦ ωN ◦ γ) for some P ∈ C[X] and we have that f(τ0), f ◦ωN ◦γ(τ0) ∈ C, the order of f(τ0) as a zero of ΨΓ,N,f (X, f(τ0)) is equal to that of ￿ γ(X−f ◦ωN ◦γ(τ0)). And now, this multiplicity coincides with the formula given in the statement, after identifying the elements in O(D,M) which fix τ0 with the corresponding order Λ ⊂ K by means of the embedding φ. Therefore, we only need to prove that the multiplicity of f(τ0) as a root of￿ γ(X − f ◦ ωN ◦ γ(τ0)) is equal to that of ΨΓ,N,f (X,X). For this purpose, we study lim τ→τ0 ￿ γ(f(τ)− f ◦ ωN ◦ γ(τ0)) ΨΓ,N,f (f(τ), f(τ)) = lim τ→τ0 ￿ γ(f(τ)− f ◦ ωN ◦ γ(τ0))￿ γ(f(τ)− f ◦ ωN ◦ γ(τ)) = lim τ→τ0 ￿ γ f(τ)− f ◦ ωN ◦ γ(τ0) f(τ)− f ◦ ωN ◦ γ(τ) . Now, if f ◦ωN ◦γ(τ0) ￿= f(τ0), the corresponding quotient is clearly 1. To study the case when f ◦ ωN ◦ γ(τ0) = f(τ0), we consider a local parameter around τ0, q(z) = z−τ0z−τ0 , and the corresponding expansion f(q) = f(τ0) + aeq e + . . . where e is the order of the point τ0 and ae ∈ C∗. Then, q ◦ ωN ◦ γ = aq for a certain a ∈ C∗, and therefore the value of the corresponding quotient is aeae−aeae ∈ C∗, since ae = 1 would imply that f ◦ ωN ◦ γ = f . Finally, if any of the values f ◦ωN ◦ γ(τ0) is not finite, we can always reduce to the previous case applying a suitable homographic transformation to f so that the values of the new Hauptmodul satisfy the required condition. 2.1.2 Singularities of the kroneckerian polynomials of prime power level We are also interested in describing, as precisely as we can, the set of singularities of the kroneckerian polynomial of a given prime power level N = pr attached to a certain Hauptmodul of a genus 0 Shimura curve (including the case of Atkin- Lehner quotients of genus 0). This result can also be used to describe the set of non-cuspidal singularities of the classical modular polynomials. First of all we are going to characterize the set of singularities and, in particular, we will obtain that it is contained in the set of complex multiplication points attached to rings containing non-trivial elements of suitable norms. The rest of the proof will deal with giving an explicit description of this set of points. If we were only interested in the prime level case, then the other inclusion would hold and 2.1 The polynomials ΨΓ,N,f 41 everything could be proved in a rather straightforward way (cf. [Gro87] for the modular case). Moreover, we are going to provide formulas for the multiplicity, in a certain sense, of each of these singularities. Lemma 2.2. Let X(D,M)W be the Atkin-Lehner quotient by W ⊂ WDM of the Shimura curve of discriminant D and level M and assume that it has genus 0. Let us denote as well the corresponding Eichler order of level M by O and fix an arbitrary integer N > 1 coprime to DM . Fix an Atkin-Lehner involution ωN and consider the kroneckerian polynomial attached to a certain Hauptmodul f of X(D,M)WC , Ψ(X,Y ) = ΨΓ,N,f (X,Y ) ∈ C[X,Y ]. A point of the curve Ψ(X,Y ) = 0 is singular if and only if it is of the form (f(P ), f(wNP )) for P ∈ H such that γ0 ωNγ1ωNωsP = P , where γ0, γ1 ∈ Γ(D,M) ￿ Γ(D,MN) and ωs ∈ W . In particular, P gives a CM point in X(D,MN) by an order Λ containing an element of norm sN2 not in NΛ, for some s ∈ S. Proof. Every point of the curve Ψ(X,Y ) = 0 is of the form (f(P ), f(wNP )) for a certain P ∈ H. Moreover, since X(D,MN)WC is non-singular and its set of com- plex points can be identified with ￿Γ(D,MN),W ￿\H, (f(P ), f(wNP )) is sin- gular if and only if there exists P ￿ ∈ H such that P ￿= P ￿ in ￿Γ(D,MN),W ￿\H and (f(P ), f(wNP )) = (f(P ￿), f(wNP ￿)), which translates into P = γ0P ￿ and ωNP ￿ = γ1ωNωsP for certain elements γ0, γ1 ∈ Γ(D,M) ￿ Γ(D,MN) and ws ∈ W . Therefore we obtain the condition in the statement of the proposi- tion. Definition. With the notations of the previous lemma, we define the multiplic- ity of Ψ at (P,ωNP ), mult(P,ωNP )Ψ, as #{P ￿ ∈ X(D,MN)W (C) : f(P ￿) = f(P ), f(ωNP ￿) = f(ωN (P ))}, i.e. the number of branches of the curve Ψ(X,Y ) = 0 through (f(P ), f(ωNP )). The first result we will prove relates this notion of multiplicity to a certain order of vanishing. Theorem 2.2. Let X(D,M)W be a genus 0 curve with a Hauptmodul f . Denote as well by O the corresponding order and Γ = ￿Γ(D,M),W ￿ the corresponding group. For a positive integer N coprime to DM fix an Eichler order of level MN inside O, O(N), the corresponding group Γ(N) = ￿Γ(D,MN),W ￿ ⊂ Γ, and ωN the corresponding involution of level N . Now, for a point P ∈ H, denote by ΓP = {γ ∈ Γ : γ(P ) = P} 42 Kroneckerian polynomials the isotropy of P and (Γ\O￿N )P = {Γα ∈ Γ\O￿N : α(P ) = ωN (P ) in Γ\H}. Then, (1) (ΓP ∩ Γ(N))\ΓP acts on the right on (Γ\O￿N )P , as follows ·[￿] : (Γ\O￿N )P → (Γ\O￿N )P Γα ￿→ Γα￿. Moreover, this action is free, i.e., if [￿] ￿= [1], then ·[￿] fixes no element. (2) The branches of the polynomial Ψ(X,Y ) = ΨΓ,N,f (X,Y ) through (P,ωNP ) are in bijective correspondence with the orbits of this action. The elements in a fixed orbit, αi, give the elements (f, f ◦ αi) which define this branch. In particular, ordf(ωNP )Ψ(f(P ), X) = e1(P ) eN (P ) mult(P,ωN (P ))Ψ(X,Y ), where ea(P ) denotes the elliptic order of the point P in the X(D,Ma)W . Proof. First of all, consider the right action by multiplication of ΓP on (Γ\O￿N )P , which is clearly well defined. Let us now prove that this action factors through (ΓP ∩Γ(N))\ΓP . Observe that any class in (Γ\O￿N )P can be written as Γα for a certain α ∈ O￿N such that α(P ) = ωN (P ) in H. Hence, consider Γα ∈ (Γ\O￿N )P with α(P ) = ωN (P ). Then, if γ ∈ ΓP , γ and ω−1N α are two elements fixing P and therefore they commute. Consider ω−1N α and ω −1 N αγ = ω −1 N α ￿. We will see that α￿ ∈ Γα if and only if γ ∈ Γ(N), proving in this way that it factors through a free action. Observe that γ = (ω−1N α) −1ω−1N α ￿ = ω−1N α ￿(ω−1N α) −1 = ω−1N α ￿α−1ωN . Hence, we obtain that ω−1N αγα −1ωN = γ ∈ Γ. Thus, if γ ∈ Γ(N), we obtain that αγα−1 ∈ Γ(N) and therefore, αγ ∈ Γ(N)α as desired. Otherwise, if γ ∈ Γ￿ Γ(N), then it follows from the above equality that αγα−1 /∈ Γ and therefore αγ /∈ Γα which proves the equivalence. In order to relate this expression with the branches through (P,ωNP ), note that the branches through P can all be parametrized as (f, f ◦ωN ◦γ) for those γ 2.1 The polynomials ΨΓ,N,f 43 such that ωNγ(P ) = ωN (P ). Moreover, for any elliptic transformation ρ fixing P , (f(ρz), f(ωNγρz)) = (f(z), f(ωNγρz)) and therefore the branch attached to ωNγρ coincides with the branch attached to ωNγ. Reciprocally, if the branches (f, f ◦ωN ) and (f, f ◦ωNγ) coincide, this means that there exists a non-constant analytic map h around P and fixing P such that (f, f ◦ ωN ) = (f ◦ h, f ◦ ωNγh). In particular, h = ρ is an elliptic transformation fixing P . Thus, the bijection between branches and orbits is now clear. Moreover, the elements in the orbit correspond exactly to the trans- formations γi ∈ Γ(N)\Γ such that (f, f ◦ωNγi) parametrizes the corresponding branch. Therefore, #(Γ\O￿N )P = #((ΓP ∩ Γ(N))\ΓP )mult(P,ωN (P ))Ψ(X,Y ) = e1(P ) eN (P ) mult(P,ωN (P ))Ψ(X,Y ). On the other hand, #(Γ\O￿N )P = ordf(ωNP )Ψ(f(P ), X) is automatic from the definition of the set (Γ\O￿N )P . Remark. Observe that mult(P,ωN (P ))Ψ(X,Y ) ≤ min(ordf(ωNP )Ψ(f(P ), X), ordf(P )Ψ(f(ωNP ), X)), and the equality holds, for example, if N is the power of an odd prime (since in this case eN (P ) = min(e1(P ), e1(ωNP ))). Theorem 2.3. Assume that M is either squarefree or a prime power. Let Y (1) be the curve X(D,M)W attached to the Eichler order O of level M and the Atkin-Lehner subgroup W . Let N = pr, for an odd prime p coprime to DM and some integer r ≥ 1. Fix Eichler orders O ⊃ O(p) ⊃ · · · ⊃ O(pr−1) ⊃ O(pr) and for any 0 ≤ k ≤ r denote by Y (pk) the corresponding curve X(D,Mpk)W . Write S = {s : ωs ∈ W}. Given 1 ≤ t ≤ r, consider Zt = ￿t k=0 ωptZt,k, where Zt,k denotes the set of CM points in Y (pt) corresponding to conjugation classes by Γ(D,Mpt) of optimal embeddings φ of Λ into O(pk) such that: • p ￿ cond(Λ), or, equivalently, condp(Λ) = 1, where condp denotes the p-part of the conductor. • If t > k ≥ 0, then φ(Λ) ￿⊂ O(pk+1), or, equivalently, φ(Λ) ∩ O(pk+1) = Z+ pφ(Λ). 44 Kroneckerian polynomials • At least one of the following two conditions holds (cf. Proposition 2.3): – The point in Y (pk) corresponding to φ is elliptic. – There exists an element λ ∈ Λ￿ of norm sp2k, for a certain s ∈ S, such that λ2 ∈ sΛ. Moreover, if s is not squarefree, wsφ(λ) ∈ sO. We will denote by Z1t,k the subset of elements for which an element of norm sp2k as above exists. Consider as well the following natural projections: πpν ,pµ : Y (pν) → Y (pµ) for integers ν ≥ µ. Then, the set of points of Y (pr) giving rise to singularities of the kronecke- rian polynomial Ψpr (X,Y ) is a subset of r￿ t=1 π−1pr,ptZt. Given P ∈ π−1pr,ptωptP0, with P0 ∈ Zt,k a CM point in the previous set, denote by Λ,Λ0 and Λ1 the orders corresponding to the CM points P0, P and ωprP in Y (1), respectively. Then, we have a ·mult(P,ωprP )Ψpr (X,Y ) = A, where A =  2e1(ωptP ), if k > 0, P0 ∈ Z1t,k and condp(Λ1) = pr−t, 1, if k = 0 and condp(Λ1) < pr−t, e1(ωptP ), otherwise, a = max(e1(P ), e1(ωprP )), e1(Q) denotes the elliptic order of the point Q in the curve Y (1) and mult(P,ωprP )Ψpr (X,Y ) denotes the number of branches of the curve through the point (P,ωprP ). Moreover, all the branches at a singular point have different tangents if and only if e1(P ) = e1(ωprP ). Otherwise, all branches are tangent at P . Remark. In Proposition 4.1 we will see that the restriction on M is not an actual restriction since all curves X(D,M)W of genus 0 satisfy this condition. Proof. Given a positive integer k, we will denote Γ(pk) = Γ(D,Mpk) and Γ(pk)￿ = Γ(D,Mpk)￿ Γ(D,Mpk+1). 2.1 The polynomials ΨΓ,N,f 45 To begin with, let us prove that any singular point can be obtained as predicted by the statement. Consider P ∈ H giving a CM point in Y (pr) and assume it is fixed by an element of the form γ0 ωNγ1ωNωs with γ1 ∈ Γ(ps1)￿, γ0 ∈ Γ(ps0)￿, s ∈ S and 0 ≤ s0, s1 ≤ r. Then, using Proposition 2.2, we obtain that the image of P in Y (1) is a CM point by an order Λ0 such that the p-part of its conductor is ps0 , if s0 + s1 ≤ r, and pr−s1 , if s0 + s1 ≥ r. Similarly, the point ωNP gives a CM point in Y (pr) and it is fixed by the transformation ωNγ0 ωNγ1ωs corresponding to an element in O(pr). Therefore, the image of ωNP in Y (1) is a CM point by an order Λ1 such that the p-part of its conductor is ps1 , if s0 + s1 ≤ r, and pr−s0 , if s0+s1 ≥ r. Observe that the case s0+s1 ≥ r is equivalent to requiring the algebraic element corresponding to the transformation to be divisible by pr in the ring of integers. Moreover, if we denote by pf0 , pf1 the p-part of the conductor of the orders corresponding to P , Λ0, and ωNP , Λ1, in Y (1) as above, we have that f0 + f1 ≤ r. Fix P ∈ H representing a point in Y (N) such that (P,ωNP ) yields a point in Y (1) × Y (1) which is a singularity of ΨN or, equivalently, such that γ0 ωNγ1ωNωsP = P for certain γ0, γ1 ∈ Γ(1)￿ Γ(N) and s ∈ S. In particular, P is a CM point and denote by pf0 , resp. pf1 , the p-part of the conductor of the order Λ0, resp. Λ1, corresponding to the projection of the point P , resp. ωNP , to Y (1). Then, γ0 ωNγ1ωNωs corresponds to an element λ ∈ Λ0￿NΛ0 of norm sN2 for some s ∈ S and it follows from the previous paragraph that either γ0 ∈ Γ(pf0) and γ1 ∈ Γ(pf1), or γ0 ∈ Γ(pr−f1) and γ1 ∈ Γ(pr−f0). Then, we define t = r − f1, k = r − f0 − f1. In the first case, we can write γ0 ωNγ1ωNωs = p f1γ0 ωptγ ￿ 1ωptωs. Now, since γ0 ∈ Γ(pf0), we obtain that ωptP is fixed by ωptγ0 ωptγ ￿ 1ωs = p f0ωpkγ ￿ 0 ωpkγ ￿ 1ωs. If we prove that the element in O corresponding to ωpkγ￿0 ωpkγ￿1ωs does not belong to O(pk+1) (we will simply write ωpkγ￿0 ωpkγ￿1ωs /∈ O(pk+1)) whenever 46 Kroneckerian polynomials t > k, we can deduce that P ∈ π−1pr,ptωptZt,k. In order to prove this, observe that ωpk γ￿0 ωpkγ￿1ωs /∈ O(pk+1) if and only if ωpk+1ωpk γ ￿ 0 ωpkγ ￿ 1ωsωpk+1 /∈ pk+1O, and since γ￿1 ∈ Γ(1)￿, this is equivalent to ωpk γ￿0 ωpkγ￿1 /∈ ωpk+1O￿pk−1 . In order to prove this assertion, observe that ωpt γ0 ωptγ￿1 = pt−kωpk γ￿0 ωpkγ ￿ 1 and multiplying on the left by ωpk+1 , if t > k, we obtain that ωpk+1ωpt γ0 ωptγ ￿ 1 = p k+1γ2ωpt−k−1 γ0ωptγ ￿ 1 ∈ ptΓ(1)￿ωpk+1Γ(1)￿. From this, we deduce that ωpk+1ωpk γ￿0 ωpkγ￿1 ∈ pkO￿pk+1 and the result follows. Similarly, in the second case, using that f1 ≤ r − f0, we can write γ0 ωNγ1ωNωs = p f1γ0 ωpr−f1γ ￿ 1ωpr−f1ωs, with γ￿1 ∈ Γ(pr−f0−f1) and since γ0 ∈ Γ(pr−f1), we obtain that P ∈ π−1 pr,pr−f1ωpr−f1Zr−f1,r−f1−f0 = π −1 pr,ptωptZt,k. Now, let us prove the formula providing the multiplicity of the singularities. If we assume that a point P ∈ H corresponding to a CM point in Y (pr) is fixed by a transformation which can be written as γ0 ωNγ1ωNωs, then for any other transformation γ0 ωNρ1ωNωs￿ , multiplying one of them by the conjugate of the other, we obtain a third transformation ωs￿ωN ρ1γ1ωNωs. Therefore we can conclude that either both elements are equal, or ωNP is elliptic. In any case, there are e1(ωprP ) such transformations. Moreover, we have seen in the previous theorem that the branch attached to ωNγ￿ coincides with the branch attached to ωNγ if and only if they are related by an elliptic transformation fix- ing P . Therefore, we obtain e1(P )/epr (P ) additional transformations for any of the transformations above, where epr (P ) denotes the order of the point P in the curve Y (pr). As a result, we obtain the number of different transformations giv- ing the same branch, namely, e1(P )e1(ωprP )/epr (P ) = max(e1(P ), e1(ωprP )). In the first part of the proof we have seen that any transformation of the form γ0 ωprγ1ωprωs fixing P can be obtained from a transformation ρ0 ωpkρ1ωpkωs fixing P0. Now, we are going to study when we can go the other way around. First of all, assume that we start with an element of the form ρ0 ωpkρ1ωpkωs, k ≥ 0, which is primitive in the ring of integers. This gives an element of the form γ￿0ωptγ ￿ 1ωptωs fixing P . Now, if we can write pr−tγ￿0ωptγ￿1ωptωs = γ0ωprγ1ωprωs 2.1 The polynomials ΨΓ,N,f 47 it will follow from the first part of the proof that Λ1 has p-conductor pr−t. Let us now prove that, if the p-conductor of Λ1 is pr−t, then a primitive element like the one we started from lifts to an element as desired. In order to do that, we will first of all study when pr−tγ￿0ωptγ￿1ωptωs can be written as γ0 ωprγ1ωprωs. This is always true if t = r. Otherwise, if t < r, pr−tγ￿0ωptγ ￿ 1ωptωs = γ0 ωprγ1ωprωs holds for certain γ0, γ1 if and only if, multiplying both sides on the right by ωpr , p−tγ￿0ωptγ ￿ 1ωptωpr is primitive in O, which holds if and only if γ￿0ωptγ￿1ωpt cannot be written as γ￿0ωpt−1γ ￿￿ 1ωpt+1 . Thus, we only need to prove that, if P ∈ Y (pr) is fixed by γ￿0ωpt−1γ￿￿1ωpt+1ωs, then the p-part of the conductor of Λ1 is strictly smaller than pr−t. Observe that since this element is not divisible by p in O, γ￿￿1 /∈ Γ(p). If we consider ωpt+1γ ￿ 0ωpt−1γ ￿￿ 1ωpt+1ωsωpt+1 = p t+1ωpt+1γ ￿ 0ωpt−1γ ￿￿ 1ωs, since γ￿￿1 /∈ Γ(p) we obtain that ωpt+1P is a CM point in Y (1) by a p-maximal order. Therefore, since ωpt+1P and ωprP are related by a transformation at- tached to an element of norm pr−t−1, the p-part of the conductor of Λ1 is at most pr−t−1. Secondly, assume now that we start from an element of the form ωptγ￿1ωptωs fixing P with γ￿1 ∈ Γ(pk), k > 0. We are going to prove that all these elements can be written as pr−tωptγ￿1ωptωs = γ0 ωprγ1ωprωs. As in the previous case, the only elements for whom this does not happen are those which can be written as ωptγ ￿ 1ωptωs = ωpt−1γ ￿￿ 1ωpt+1ωs. In particular, if t = k = 1, the condition translates into ωpγ ￿ 1ωpωs = γ ￿￿ 1ωp2ωs. Since γ￿1 ∈ Γ(p), the lefthand side is not primitive in O, while the righthand side is. Therefore we reach a contradiction. Otherwise, if t > 1, we can write ωptγ￿1ωptωs = pωpt−1γ￿￿1ωpt−1ωs. Therefore this element does not lift in the 48 Kroneckerian polynomials desired fashion if and only if ωpt−1γ￿￿1ωpt−1ωs = ωpt−2γ￿￿￿1 ωptωs. But putting together both equalities, ωptγ ￿ 1 = pωpt−2γ ￿￿￿ 1 we have an equality between an element which is primitive in the quadratic order and another one which is not. Summing up, we have already seen that a(mult(P,ωprP )Ψpr (X,Y )− 1) coincides with the number of transformations of the form γ0 ωprγ1ωprωs fixing P . Now, if P ∈ π−1pr,ptωptZt,k, all such transformations give elements in the order corresponding to ωptP of norm s or sp2k, for some s ∈ S. It is worth noting that, since all these elements must lie in a quadratic p-maximal order, given two elements λ,λ￿ of norm in Sp2k, then p2k either divides λλ￿ or λλ￿. Hence, it follows that the transformations attached to these elements are related through an elliptic transformation. As a result, if P ∈ π−1pr,ptωptZ1t,k is such that ωprP has p-conductor pr−t, then the number of transformations coming from elements of norm in Sp2k of the form γ0 ωprγ1ωprωs is e1(ωptP ). On the other hand, the number of transformations of the same form coming from elliptic elements is as well e1(ωptP ). Therefore, if k > 0 we obtain 2e1(ωptP ) transfor- mations and if k = 0, e1(ωptP ). On the other hand, if ωprP has p-conductor strictly smaller than pr−t, none of the transformations coming from elements of norm in Sp2k rise to elements of the expected form, and thus what we obtain are e1(ωptP ) transformations if k > 0 and 1 if k = 0 (only the one corresponding to the identity). Now, we only have to find out how many of these transfor- mations do not correspond to branches, and that is those which correspond to elliptic transformations fixing P or ωprP , that is e1(P ) + e1(ωprP ) − epr (P ), which coincides with a. Thus we obtain that the number of branches through a singularity coincides with the formula given in the statement. Now, it only remains to study the tangents of the different branches through a singularity. In order to study this we will make use of expansions which will be introduced later on, when expansions are studied on their own, in Sec- tion 3.3. Denote e = e1(P ) and e￿ = e1(ωprP ). Then, the branches of the curve Ψpr (X,Y ) = 0 attached to the Hauptmodul f through P can be obtained as (f, f ◦ ωpr ◦ λ) where λ runs over transformations γ0 ωprγ1ωprωs giving the different branches which we have computed before. Since f is a Hauptmodul, 2.1 The polynomials ΨΓ,N,f 49 we can choose a local parameter of the form q = κ z−P z−P for a certain κ ∈ C∗ such that f = a0 + qe + o(qe). Then, with this same parameter, we can consider an expansion of f ◦ ωpr around P . Since it is a Hauptmodul for the curve corre- sponding to the quaternion order w−1pr Owpr and for which P is an elliptic point of order e￿, then f ◦ ωpr = b0 + b1qe￿ + o(qe￿) with b1 ∈ C∗. Now, if α is the element in the quadratic order corresponding to the transformation λ fixing P , q ◦ λ = Normα α2 q = α α q (cf. Proposition 3.6). Observe that, since α lies in a quadratic field but not in Q, this constant factor is at most quadratic and different from 1. Therefore, if it is a root of unity, it must be primitive of order 2, 3, 4 or 6. The only primitive elements for which this holds are α ∈ {1 ± i, ±3± √−3 2 } ∪ { √−D} for positive squarefree integers D > 1, which have norm 2, 3, D, respectively, and none of these has the form sp2k for a prime p, s ∈ S and some k > 0. Now, we will deal with the different cases one by one. If e1(ωprP ) > 1, whenever we have a singularity it comes from a primitive element λ of norm sp2k, k > 0, and we have two branches through P , which can be parametrized as follows: (f, f ◦ ωpr ) = (a0 + qe + . . . , b0 + b1qe￿ + . . . ), (f, f ◦ ωprγ) = (a0 + qe + . . . , b0 + b1 λ e￿ λe￿ q e￿ + . . . ). Therefore, if e = 1, both have common tangent Y = 0 and, if e = e￿, the tangents are different, b1X − Y = 0 and b1 λ e λeX − Y = 0, respectively. The case e > 1, e￿ = 1 is totally analogous to the case e = 1, e￿ > 1 reversing the role of X and Y . Finally, in the case e = e￿ = 1, f = a0+q+· · · and f ◦ωpr = b0+b1q+· · · and every non-trivial transformation fixing P changes q non-trivially. Remark. It is worth noting that, if either P or ωprP are elliptic, then there are at most two branches through (P,ωprP ). Moreover, if both are elliptic, then these two branches have different tangents. Even though this is not in general the only type of points related to elliptic points having these properties, we will exploit this fact later on. Corollary 2.4. The condition condp(Λ1) < pr−t in the previous theorem can be rewritten as: 50 Kroneckerian polynomials • e1(ωptP ) = e1(ωpt+1P ) > 1, or • e1(ωptP ) = 1 and mult(P,ωpt+1P )Ψpt−1 > mult(P,ωptP )Ψpt−2 −mult(P,ωpt+1P )Ψpt−3 . Recall that Ψ1(X,Y ) = X − Y , and we simply put Ψu(X,Y ) = 1 if u is not a positive integer. Proof. In order to prove the equivalence, recall that, as we have seen in The- orem 2.3, the condition condp(Λ1) < pr−t is equivalent to the existence of an element of the form γ0 ωpt−1γ1ωpt+1ωs fixing P with γ1 /∈ Γ(p). First, let us assume that condp(Λ1) < pr−t. Then, there exists a trans- formation of the form γ0 ωpt−1γ1ωpt+1ωs fixing P with γ1 /∈ Γ(p). Therefore, ωpt+1γ0 ωpt−1γ1ωs is an element fixing ωpt+1P . Since γ1 /∈ Γ(p), we obtain that the quadratic order corresponding to ωpt+1P has conductor coprime to p, and therefore it coincides with the order attached to ωptP . Since both of these points are related by a transformation coming from an element in O, we can conclude that, if ωptP is elliptic, so is ωpt+1P . Reciprocally, if e1(ωptP ) = e1(ωpt+1P ) > 1, we obtain that the conductor of ωprP is divisible at most by pr−t−1. Otherwise, in the case e1(ωptP ) = 1, it will suffice to prove that the number of elements of the form γ0 ωpt−1γ1ωpt+1ωs fixing P with γ1 /∈ Γ(p) is mult(P,ωpt+1P )Ψpt−1 − (mult(P,ωptP )Ψpt−2 −mult(P,ωpt+1P )Ψpt−3). Consider the set {γ0 ωpt−1γ1ωpt+1ωs : it fixes P and γ1 /∈ Γ(p)}. Even though we will not make use of it, it is easy to see that this set has cardinality either 0 or 1. Now, if we consider the bigger set {γ0 ωpt−1γ1ωpt+1ωs : it fixes P}, since we are in the non-elliptic case, it has cardinality mult(P,ωpt+1P )Ψpt−1 . Moreover, the two sets differ by {γ0 ωpt−1γ1ωpt+1ωs : it fixes P and γ1 ∈ Γ(p)}, and every such element can be written as γ0 ωpt−2γ1ωptωs. 2.2 Kroneckerian polynomials for the Hauptmoduln tD of X(D, 1) 51 The number of such elements is mult(P,ωptP )Ψpt−2 . However, not all these el- ements come from one of the form γ0 ωpt−1γ1ωpt+1ωs. The set of those for whom it is not possible, as we have seen in the proof of Theorem 2.3, is {γ0 ωpt−3γ1ωpt+1ωs : it fixes P}, which has cardinality mult(P,ωpt+1P )Ψpt−3 . As a result, the expression above follows. 2.2 Kroneckerian polynomials for the Hauptmod- uln tD of X(D, 1) In the previous section, we dealt with the case of curves X(D,M)W of genus 0. Now we are going to restrict our study to the case M = 1,W = {id}, which amounts to D = 6, 10, 22. In this situation, these are genus 0 curves with no real points and therefore, there exist no Hauptmoduln defined over Q. In the previous section we studied the properties satisfied by a kroneckerian polynomial attached to a generic Hauptmodul defined over a generic number field. Now we are going to introduce some particular Hauptmoduln for all these curves and prove that the corresponding kroneckerian polynomials are defined over Q even though the Hauptmoduln are not, together with some additional arithmetic properties concerning their symmetries and their reduction. First of all, we are going to introduce the Hauptmodul tD, which can be seen as an extension to the cases D = 10, 22 of the construction given in [BT07b] for the case D = 6 (cf. [Kur79], [Jor81]). Similarly, we could reproduce the construction given in this section to Atkin-Lehner quotients X(D,M)W for a certain subgroup W ⊂WDM of index at least 4 such that X(D,M)W has genus 0 and no rational points, which is the case, for example for D = 6, N = 5 and W = {ω2} or W = {ω10}. Theorem 2.4. Fix D = 6, 10, 22 and for each of these values of D consider the prime integer c(D) =  3, D = 6, 2, D = 10, 11, D = 22. Then, there exists a Hauptmodul tD for the curve X(D, 1) satisfying the following properties: (1) Attached to an element of norm −D in O, there is a hyperbolic symmetry 52 Kroneckerian polynomials s on the upper half-plane such that, for every z ∈ H, tD(z) = tD(s(z)). (2) Let α1,α2,α3 be the three non-trivial involutions of X(D, 1) in WD, then {tD ◦ αj}j=1,2,3 = {−tD, t−1D ,−t−1D }. More precisely, D α1 tD ◦ α1 α2 tD ◦ α2 α3 tD ◦ α3 6 ω2 −t6 ω3 t−16 ω6 −t−16 10 ω2 t −1 10 ω5 −t10 ω10 −t−110 22 ω2 −t22 ω11 t−122 ω22 −t−122 (3) The function tD is a Hauptmodul over the field Q(i, ￿ c(D)) for the curve X(D, 1). (4) The canonical model X(D, 1) is given by a function jD : H→ P2(C) which, in projective coordinates, can be written as jD = (uD,1 : uD,2 : 1), where its coordinate functions relate to tD as follows: uD,1 = ￿ −c(D) t 2 D + 1 2tD , uD,2 = ￿ c(D) t2D − 1 2tD . Moreover, they satisfy the equation u2D,1 + u2D,2 + c(D) = 0. (5) The functions uD,1, uD,2 and uD,3 = uD,1/uD,2 are Hauptmoduln defined over Q for the three quotients of X(D, 1) by a non-trivial Atkin-Lehner involution: D uD,1 uD,2 uD,3 6 ω3 ω6 ω2 10 ω2 ω10 ω5 22 ω11 ω22 ω2 (6) A Hauptmodul defined over Q for the quotient X(D, 1)WD = X(D, 1)+ is 2.2 Kroneckerian polynomials for the Hauptmoduln tD of X(D, 1) 53 given by t+D such that D u2D,1 u 2 D,2 u 2 D,3 6 3t + 6 1−t+6 3 t+6 −1 −t+6 10 −2t+10 2(t+10 − 1) t + 10 1−t+10 22 11t + 22 1−t+22 11 t+22−1 −t+22 Proof. For the moment, let us denote X = X(D, 1), X1, X2, X3 the quotient curves corresponding to α1,α2,α3 respectively and X+ the double quotient curve. The Riemann-Hurwitz formula applied to the Galois cover X → X+ yields that this morphism is ramified exactly at 3 points x1, x2, x3 of X+ with ramification index 2, each of them corresponding respectively to the projection to X+ of the ramification divisors of X → Xi. Since the morphisms αi are defined over Q, we obtain therefore that x1, x2, x3 ∈ X+(Q). Now we can consider a Hauptmodul t+ : X+(C)→ P1(C) such that t+(x1) = 1, t+(x2) =∞, t+(x3) = 0, which is therefore defined over Q. Similarly, xj ∈ Xl(Q) if 1 ≤ j, l ≤ 3, j ￿= l, and therefore Xl ∼= P 1Q, 1 ≤ l ≤ 3. Therefore, there exist Hauptmoduln u˜l for Xl defined over Q. Now we are going to give expressions for a certain choice of these u˜l in terms of t+. The morphism X1 → X+ has degree 2 and ramifies exactly at x2, x3, which are rational points of X+ where t+ takes the values ∞ and 0, respectively. Thus, we can choose u˜1 ∈ Q(X1) such that u˜21 = k1t+ for a suitable squarefree integer k1 which we are going to determine next. Then, since Q(t+) ⊂ Q(u˜1), we will automatically have that Q(X1) = Q(u˜1). Similarly, u˜2, u˜3 can be taken such that u˜22 = k2 t+ − 1 t+ and u˜23 = k3(t+ − 1). Note that we can take u˜3 = u˜1u˜2, that is k3 = k1k2. It is worth mentioning as well that k3u˜21−k1u˜23−k1k3 = 0 and, therefore, this pair of functions provides and affine chart for a canonical model of the Shimura curve X. In particular, sinceX(R) = ∅, we obtain that k1, k2 < 0 and k3 > 0. By definition, u˜l◦αl = u˜l and u˜l ◦ αj = −u˜l for j ￿= l. Moreover, we can compute the value of these 54 Kroneckerian polynomials functions at the points xj : x1 x2 x3 u˜1 ± √ k1 ∞ 0 u˜2 0 ± √ k2 ∞ u˜3 0 ∞ ± √−k3 Let us now discuss the three cases separately: • D = 6: Choose α1 = ω2, α2 = ω3 and α3 = ω6. It follows from [Ogg83], cf. Theorem 4.1, that x1 ∈ CM(Z[i]), x2 ∈ CM(Z[ 1+ √−3 2 ]), x3 ∈ CM(Z[ √−6]). Therefore, u˜l(x1) ∈ HCF (Q(i)) = Q(i), u˜l(x1) ∈ HCF (Q( √−3)) = Q(√−3), which taking into account the restriction on the signs of the ki’s we have observed before, yields that k1 = −1, k2 = −3, k3 = 3. Define u6,1 = 3/u˜2 and u6,2 = 3/u˜3. • D = 10: Choose α1 = ω2, α2 = ω5 and α3 = ω10. As in the previous case, we obtain that x1 ∈ CM(Z[ √−2]), x2 ∈ CM(Z[ √−5]), x3 ∈ CM(Z[ √−10]). Therefore, u˜l(x1) ∈ HCF (Q( √−2)) = Q(√−2), u˜l(x2) ∈ HCF (Q( √−5)) = Q(i,√−5), which having the signs of the ki’s in mind yields that k1 = −2, k2 = −1,−5, k3 = 2, 10. Observe, however, that if we had k1 = −2, k2 = −5, k3 = 10, then the values u˜1(x3) = 0, u3(x3) = ± √−10 would not generate the Hilbert Class Field HCF (Q( √−10)) = Q(√−10,√−2) over Q(√−10) in contradiction with the properties of the canonical model. Therefore k1 = −2, k2 = −1, k3 = 2. Define u10,1 = u˜1 and u10,2 = u˜3. 2.2 Kroneckerian polynomials for the Hauptmoduln tD of X(D, 1) 55 • D = 22: This case is totally analogous to the case D = 6. In this case k1 = −1, k2 = −11, k3 = 11 and u22,1 = 11/u˜2 and u22,2 = 11/u˜3. We can now define the Hauptmodul tD = ￿ c(D) iuD,1 + uD,2 = iuD,1 − uD,2￿ c(D) , which is clearly defined overQ(i, ￿ c(D)). With these definitions, the statements (3), (4) and (5) are now immediate and it only remains to check that the first two hold too. It follows from [Shi75b] that for any f ∈ Q(X(D, 1)), if we consider the symmetry attached to an element of norm −1 in O(D, 1), then f ◦ s = f . Therefore, if we consider the symmetry s ◦ ωD corresponding to an element of norm −D, from the values of the corresponding functions at the points x1, x2, x3, we obtain that uD,1 ◦ (s ◦ ωD) = −uD,1 and uD,2 ◦ (s ◦ ωD) = uD,2. Now, (1) follows trivially. Finally, in order to prove (2), we observe that uD,j ◦ωl = ±uD,j . Then, using the explicit description of tD in terms of uD,j and that u2D,1 + u2D,2 + c(D) = 0, the result follows. These properties of tD will allow us to prove more properties of the corre- sponding kroneckerian polynomials ΨΓ,N,tD (X,Y ) ∈ Q(i, ￿ c(D))[X,Y ]. Proposition 2.6. For every integer N > 1 coprime to D, the polynomial ΨΓ,N,tD (X,Y ) can be taken as a primitive polynomial in Z[X,Y ]. Also, the global degrees of all the non-zero monomials of ΨΓ,N,tD (X,Y ) are even. More- over, if we write ΨΓ,N,tD (X,Y ) = ψ(N)￿ j,k=0 ajkX jY k, aj,k ∈ Z, then the coefficients satisfy the following symmetry conditions: ajk = akj = aψ(N)−k,ψ(N)−j = aψ(N)−j,ψ(N)−k. Proof. The symmetry property of tD given in Theorem 2.4 is equivalent to the following symmetry properties of uD,1 and uD,2: uD,1(s(z)) = −uD,1(z), uD,2(s(z)) = uD,2(z). 56 Kroneckerian polynomials If we define u = iuD,1 + uD,2 = ￿ c(D) tD , then u is a Hauptmodul for X(D, 1) defined over Q(i). Therefore, for every positive integer N coprime to D, we can consider ΨΓ,N,u(X,Y ) ∈ Z[i][X,Y ] a kroneckerian polynomial attached to the Hauptmodul u. Recall that ΨΓ,N,u(X,u) is irreducible over Q(i)(u), and has u ◦ ωN as a root. Then, for all z ∈ H, 0 = ΨΓ,N,u(u(z), u(ωNz)) = ΨΓ,N,u(u(z), u(ωNz)); and, taking sz for z, 0 = ΨΓ,N,u(u(sz), u(ωNsz)). We note that u(s(z)) = u(z) and u(ωNs(z)) = u(s(s−1ωNs)(z)) = u((s−1ωNs)z). Since the element s−1wNs has norm N and is primitive, it belongs to O1wNγ for some γ ∈ O1. Being u a Γ-automorphic function, for every z ∈ H, we have u((s−1ωNs)z) = u(ωNγz) and 0 = ΨΓ,N,u(u(z), u(ωNγz)), which applying the map z ￿→ γ−1z gives 0 = ΨΓ,N,u(u(z), u(ωNz)). Since the conjugated polynomial ΨΓ,N,u(X,Y ) is irreducible, it follows that ΨΓ,N,u(X,u) is irreducible over Q(i)(u) and has u ◦ωN as a root, which implies that we can take ΨΓ,N,u(X,Y ) ∈ Z[X,Y ]. Similarly, if we choose α an Atkin-Lehner involution according to Theo- rem 2.4 so that tD ◦α = −tD for every z ∈ H, we have, in addition, the identity 0 = ΨΓ,N,u(u(αz), u(ωNαz)) = ΨΓ,N,u(−u(z),−u(ωNz)). Therefore, we obtain that ΨΓ,N,u(−X,−Y ) = ±ΨΓ,N,u(X,Y ). Observe that if ΨΓ,N,u(−X,−Y ) = −ΨΓ,N,u(X,Y ), thenΨΓ,N,u(X,−X) = −ΨΓ,N,u(−X,X) = −ΨΓ,N,u(X,−X), which yields ΨΓ,N,u(X,−X) = 0 and therefore that X +Y is a factor of ΨΓ,N,u(X,Y ) which is a contradiction because of the irreducibility, since N > 1. As a result, the global degrees of all the non-zero monomials of the polynomial ΨΓ,N,u(X,Y ) are even. Now, since tD = ￿ c(D)u−1 and the monomials of the kroneckerian polyno- mial ΨΓ,N,tD (X,Y ) are even, we can take ΨΓ,N,tD (X,Y ) ∈ Z[X,Y ]. 2.2 Kroneckerian polynomials for the Hauptmoduln tD of X(D, 1) 57 It only remains to prove that ajk = aψ(N)−j,ψ(N)−k for 0 ≤ j, k ≤ ψ(N). We consider now an involution α such that tD ◦ α = 1/tD. Replacing z by α(z) in the identity 0 = ΨΓ,N,tD (tD(z), tD(ωNz)), it follows that 0 = tD(z) ψ(N)tD(ωNz) ψ(N)ΨΓ,N,tD (1/tD(z), 1/tD(ωNz)), for every z ∈ H. Therefore, up to multiplication by ±1, the polynomial XΨ(N)Y Ψ(N)ΨΓ,N,tD (X −1, Y −1) must be the polynomial ΨΓ,N,tD (X,Y ). How- ever, if XΨ(N)Y Ψ(N)ΨΓ,N,tD (X−1, Y −1) = −ΨΓ,N,tD (X,Y ), we would have ΨΓ,N,tD (X −1, X) = −ΨΓ,N,tD (X,X−1) which yields that ΨΓ,N,tD (X,X−1) = 0 and therefore that XY − 1 is a factor of ΨΓ,N,tD (X,Y ) which, as before, is a contradiciton. This gives the claimed relation between the coefficients of ΨΓ,N,tD (X,Y ). From now on, we will fix a canonical model for X(D, 1), jD, and we will restrict ourselves to the case N = p ￿ D. In the analogous situation when D = 1, there is a well-known formula for the reduction modulo p of the kro- neckerian polynomial of level p, namely the Kronecker congruence formula. Its analog in the case of Shimura curves is the Eichler-Shimura congruence formula, introduced in Theorem 1.19. We will deduce an analog to the Kronecker congruence formula for the re- duction of the polynomial ΨΓ,p,tD , from the Shimura reciprocity law. We will first prove some auxiliary results. Lemma 2.3. Let HD be an indefinite rational quaternion algebra of discrimi- nant D and p ￿ D be a prime number. Then, for any sufficiently large prime r, there exists a quadratic imaginary field K satisfying: (a) p decomposes in K, (p) = pp￿; (b) K splits HD; (c) The residual degree of p in the extension K(1)|K is r, where K(1) denotes the Hilbert Class Field of K. Proof. Let S0 be a finite set of primes containing all positive primes smaller than 4p, not dividing D and different from p. If p ≡ 1 (mod 4), assume further that S0 contains a prime q ￿ D such that ￿ p q ￿ = ￿ q p ￿ = −1. Define α0 = ￿ p∈S0 p and let r ≥ 3 be a prime such that 4Dα0 < pr/2. 58 Kroneckerian polynomials For any prime q | D, define an integer 0 ≤ Aq < q if q ￿= 2 and A2 ≤ 4 if q = 2 | D such that: • In the case q = 2, when 2 | D: – if p ≡ ±3 (mod 8), then A2 = 4; – otherwise, A2 = 2; • In the case q ￿= 2, q | D, we define Aq such that A2q − pr is not a square modulo q. This can be done, for example, in the following way: – if −p is not a square modulo q, then Aq = 0; – if −p is a square modulo q, −p ≡ b2 (mod q), let aq be an integer such that a2q + 1 is not a square modulo q. Such an integer exists, otherwise the set of squares would be invariant by adding 1, which would mean that it is the whole Fq. Now define Aq ≡ braq (mod q) which satisfies the desired condition. In this situation, the Chinese Remainder Theorem grants the existence of an integer 0 < A ≤ 4D such that:￿ α0A ≡ Aq (mod q), for q | D, q ￿= 2, and α0A ≡ A2 (mod 8), for 2 | D. Now define α = Aα0 and consider integers β, d with d squarefree such that β2d = α2 − pr. Since α < pr/2, we obtain that d < 0. Since, by construction, there exists a prime in S such that −d is not a square modulo this prime (or modulo 4 if this prime is 2), then d ￿= −1; moreover, it is not divisible by any prime in S0 ∪ {q : q | D} ∪ {p}. Therefore −d > 4p. Consider the quadratic imaginary field K = Q( √ d), whose discriminant DK is either d or 4d depending on d (mod 4). We are going to prove that this quadratic field K together with the prime r satisfy the properties we were looking for. (a) Consider the ideal (p,α + β √ d) with α,β those given above. Then, since gcd(p, 2α) = 1, (p) = (p,α+β √ d)(p,α−β√d). Therefore, p is decomposed in K. Let p = (p,α+ β √ d). (b) K splits HD if and only if the Kronecker symbols ￿ DK q ￿ are not 1, for any q | D. This follows immediately by the choice of α. 2.2 Kroneckerian polynomials for the Hauptmoduln tD of X(D, 1) 59 (c) The Hilbert Class Field K(1)|K is the class field corresponding to the ideal class group of K, C(OK). In C(OK) we consider the class of the ideal p. First of all, observe that pr ⊆ (pr,α + β√d) = (α + β√d) and because both ideals have the same norm we conclude that both sides are equal, and therefore pr is principal. Since r is prime, we conclude that the order of p in the class group is either 1 or r. But there are no elements of norm p in OK , because −d > 4p. Thus, p has order r in ClK , and by the well-known decomposition theorem of class field theory (cf. [Neu99]), this order is equal to the residual degree of p as we wanted. Lemma 2.4. Let p,R be odd primes such that R > (p + 1)2 and consider Ψ(X,Y ) ∈ Zp[X,Y ] a primitive symmetric polynomial of degree p + 1 in each variable. Assume further that there exists α ∈ FpR ￿Fp such that Ψ(α,αp) = 0. Then, there exists a ∈ Z￿ pZ such that Ψ(X,Y ) ≡ a(X − Y p)(Y −Xp) (mod p). Proof. Let Ψ(X,Y ) = ￿ 0≤i,j≤p+1 ai,jX iY j , with ai,j = aj,i. Then, 0 = Ψ(α,αp) = ￿ 0≤i,j≤p+1 ai,jα i+jp = (p+1)2￿ k=0 ￿ ￿ 0≤i,j≤p+1 i+jp=k ai,j ￿ αk. Define, therefore, ci,j = ￿ 0≤i,j≤p+1 i+jp=k ai,j ∈ Zp. Since α ∈ FpR￿Fp and R > (p+1)2 is prime, we have that {1,α,α2, . . . ,α(p+1)2} are linearly independent over Fp and therefore all these coefficients must be 0 modulo p. Observe, that if k ￿≡ 0, 1 (mod p), there exists only one coefficient ai,j with i+ jp = k, namely, that with (i, j) corresponding to the representation of the integer k in base p. Similarly, if k = 0, 1, (p+1)2−1, (p+1)2, the same is true. Therefore, ai,j ≡ 0 (mod p) if i ￿≡ 0, 1 (mod p) and a0,0, a1,0, ap,p+1, ap+1,p+1 are 0 modulo p as well. Now, if k ≡ 0 (mod p), we have that cpk￿ = ap,k￿−1 + a0,k￿ . Note that a0,k￿ = ak￿,0 ≡ 0 (mod p) if k￿ ￿= p, p + 1, but if k￿ = p, cp2 = ap,p−1 + a0,p and ap,p−1 = ap−1,p ≡ 0 (mod p) which implies that a0,p is a multiple of p too. 60 Kroneckerian polynomials Therefore, only a0,p+1 can be different from 0 modulo p and it satisfies that ap,p + a0,p+1 ≡ 0 (mod p). Similarly, if k ≡ 1 (mod p), cpk￿+1 = ap+1,k￿−1 + a1,k￿ , studying the coef- ficients a1,k￿ we reach the conclusion that only a1,1 can be non-zero modulo p and it satisfies that a1,1 + ap+1,0 ≡ 0 (mod p). Therefore, all coefficients ai,j are multiples of p except for a0,p+1, a1,1, ap,p, ap+1,0 and these satisfy −a1,1 ≡ ap+1,0 ≡ a0,p+1 ≡ −ap,p (mod p) as desired. Lemma 2.5. Let p ≥ 3 be a prime, a ∈ Z∗p and r a positive integer. Let Ψ1(X,Y ),Ψ2(X,Y ) ∈ Zp[X,Y ] be primitive symmetric polynomials of degree p + 1 in each variable such that psΨ1(X + 2a/pr, Y + 2a/pr) = Ψ2(X,Y ), for a suitable positive integer s. If Ψ1(X,Y ) ≡ (X − Y p)(Y −Xp) (mod p), then Ψ2(X,Y ) ￿≡ b(X − Y p)(Y −Xp) (mod p) for any b ∈ Z￿ pZ. Proof. Assume that Ψ2(X,Y ) ≡ b(X−Y p)(Y −Xp) (mod p) and we will reach a contradiction. We will begin by computing some of the coefficients di,j of the polynomial p2rpΨ1(X + 2a/pr, Y + 2a/pr). Let us write Ψ1(X,Y ) = (X − Y p)(Y −Xp) + p ￿ 0≤i,j≤p+1 ci,jX iY j , where ci,j ∈ Zp, and then d0,0 ∈cp+1,p+1(2a)2p+2p1−2r + 2cp+1,p(2a)2p+1p1−r + (2a)2p + p1+rZp, d1,0 ∈cp+1,p+1(2a)2p+1(p+ 1)p1−r + cp+1,p(2a)2p(2p+ 1)p+ (2a)2p−1p1+r + p2+rZp, d1,1 ∈cp+1,p+1(2a)2p(p+ 1)2p+ 2cp+1,p(2a)2p−1(p+ 1)p2+r + 2cp+1,p−1(2a)2p−2(p2 − 1)p1+2r + p2+2rZp, dp,p =cp+1,p+1(2a) 2(p+ 1)2p2rp+1−2r + 2cp+1,p(2a)(p+ 1)p2rp−r+1 + p2rp + cp,pp 2rp+1, dp+1,0 ∈cp+1,p+1(2a)p+1prp−r+1 + cp+1,p(2a)pprp+1 + prp+r+1Zp. The assumption we have made implies that νp(d1,1) = νp(dp,p) = νp(dp+1,0) < νp(d0,0), νp(d1,0) 2.2 Kroneckerian polynomials for the Hauptmoduln tD of X(D, 1) 61 and the description of dp,p provides that νp(dp,p) ≥ 2r(p−1)+1 ≥ 1+2r. There- fore, p2r+1 must divide all the coefficients above. In particular, νp(cp+1,p+1) ≥ r and we define c￿p+1,p+1 = cp+1,p+1p−r. Now, each of the di,j above yields a condition on the p-adic valuation of the coefficients, namely: (i) d0,0: νp(c￿p+1,p+1(2a)2 + 2cp+1,p(2a) + pr−1) ≥ r + 1; (ii) d1,0: νp(c￿p+1,p+1(2a) + cp+1,p(2p+ 1)) ≥ r; (iii) dp+1,0: νp(c￿p+1,p+1(2a) + cp+1,p) ≥ r. Now, from (i) and (iii) we obtain that νp(cp+1,p) = r − 1 which using (ii) gives that νp(c￿p+1,p+1) = r − 1. Using this information on d1,1, it gives that νp(d1,1) = 2r < 1 + 2r and therefore we reach a contradiction. Proposition 2.7. Let X(D, 1) be the Shimura curve of discriminant D for D = 6, 10, 22 with canonical model jD : H → P2, given by the affine chart jD = (uD,1 : uD,2 : 1) such that u2D,1 + u2D,2 + c(D) = 0. Then, if p,R are odd primes, with p ￿ c(D) and R ￿ p, there exists a CM point P such that uD,1(P ), uD,2(P ) are algebraic p-integers and the reduction modulo p of any of them generates a finite field of degree R. Proof. Let uD,3 = uD,1uD,2 . For u = uD,j , j = 1, 2, 3, there exists ω = ωq, q|D, an Atkin-Lehner involution of level q such that u◦ω = −u. Let also Ψ(X,Y ) be the kroneckerian polynomial of level p, for some prime p coprime to D. Following Lemma 2.3, consider a CM point P attached to the ring of integers of a quadratic field K, where p decomposes, and such that the residual degree at p of K(1)|K is R. Fix p a prime of K above p, and from now on we will work in the p-adic completion of K, which is identified throughout with Qp. If u(P ) is a p-adic algebraic integer but its reduction modulo p lies in F∗p, then, we can write u(P ) = a + bpr, for an integer 0 < a < pr, p ￿ a, and b whose reduction lies in FpR ￿ Fp. Now, choose ω such that u ◦ ω = −u and, in particular, u(ωP ) = −a− bpr. In this way, we obtain functions u￿ = u− a pr , u￿￿ = u+ a pr , whose kroneckerian polynomials of level p, Ψu￿ ,Ψu￿￿ lie in Zp[X,Y ]. Now, Shimura reciprocity law ensures that there exists a transformation ωp of norm 62 Kroneckerian polynomials p such that u￿(ωpP ) = Frobp(u￿(P )) and u￿￿(ωpP ) = Frobp(u￿￿(P )), giving rise to points of the form (α,αp) with α ∈ FpR of the corresponding kroneckerian polynomials. Therefore, we can apply Lemma 2.4 to obtain the congruence Ψu￿(X,Y ) ≡ Ψu￿￿(X,Y ) ≡ (X − Y p)(Y −Xp) (mod p). Since u￿￿ = u￿ + 2apr , it follows that cΨu￿(X,Y ) = Ψu￿￿(X + 2apr , Y + 2a pr ) and therefore, by Lemma 2.5, we reach a contradiction. Assume now, that u(P ) is a p-adic algebraic integer whose reduction modulo p is 0. Then, one of the functions uD,1, uD,2, uD,3 is in the previous case, and therefore a contradiction is also reached. Finally, if uD,1(P ), uD,2(P ) are not p-adic integers, then uD,1(P ) = u￿D,1(P ) pr and uD,2(P ) = u￿D,2(P ) ps with u ￿ D,1(P ), u ￿ D,2(P ) ￿≡ 0 (mod p). It follows easily from u2D,1+u2D,2+c(D) = 0 that r = s and u￿D,1(P )2+u￿D,2(P )2 ≡ 0 (mod p) and therefore uD,3(P ) = u￿D,1(P ) u￿D,2(P ) must be a p-adic algebraic integer whose reduction modulo p lies in F∗p, which is not possible by the discussion above. Therefore, the point P gives us the result. Corollary 2.5. Let D = 6, 10, 22 and p a prime not dividing D. Then, ΨΓ,p,uD,j ≡ a(X − Y p)(Y −Xp) (mod p), for some a ∈ Z￿ pZ. Proof. It follows immediately after using the output of the previous proposition in Lemma 2.4. Theorem 2.5. Let D = 6, 10, 22 and p a prime not dividing D. Then, the gen- eral form of the reduction modulo p of the kroneckerian polynomial ΨΓ,p,tD (X,Y ) depends only on p (mod 4c(D)). More precisely, up to a non-zero multiplicative constant, one has that ΨΓ,p,tD (X,Y ) ≡  (X − Y p)(Y −Xp) (mod p), if p ≡ 1 (mod4) and ￿ c(D) p ￿ = 1, (X + Y p)(Y +Xp) (mod p), if p ≡ 1 (mod4) and ￿ c(D) p ￿ = −1, (1−XY p)(1− Y Xp) (mod p), if p ≡ 3 (mod4) and ￿ c(D) p ￿ = −1, (1 +XY p)(1 + Y Xp) (mod p), if p ≡ 3 (mod4) and ￿ c(D) p ￿ = 1. Proof. According to Theorem 2.4, uD,1, uD,2 are Hauptmoduln for certain quo- tient curves of X(D, 1); therefore, we can consider the kroneckerian polynomials 2.2 Kroneckerian polynomials for the Hauptmoduln tD of X(D, 1) 63 ΨΓ1,p,uD,1(X,Y ), ΨΓ2,p,uD,2(X,Y ) ∈ Z[X,Y ], such that ΨΓ1,p,uD,1(uD,1, uD,1 ◦ wp) = 0, ΨΓ2,p,uD,2(uD,2, uD,2 ◦ wp) = 0. Now, by the previous corollary, for these kroneckerian polynomials we have congruences ΨΓk,p,uD,j (X,Y ) ≡ (X − Y p)(Y −Xp) (mod p), j = 1, 2, up to a non-zero multiplicative constant. At this point, we recall the expression uD,1 = ￿−c(D) t2D + 1 2tD , and consider the ring homomorphism ϕ˜1 : Z[X,Y ] −→ Z[ ￿ −c(D), 12 ][X,X−1, Y, Y −1] defined by ϕ˜1(X) := ￿−c(D) X2 + 1 2X , ϕ˜1(Y ) := ￿−c(D) Y 2 + 1 2Y . It induces a homomorphism between the quotient rings ϕ1 : Z[X,Y ] (ΨΓ1,p,uD,1(X,Y )) −→ Z[ ￿−c(D), 12 ][X,X−1, Y, Y −1] (ΨΓ,p,tD (X,Y )) . If we fix a prime of Q( ￿−c(D)) lying above p and reduce, we obtain a ring homomorphism ϕ1 : Z/(p)[X,Y ] ((X − Y p)(Y −Xp)) −→ F[X,X−1, Y, Y −1] (ΨΓ,p,tD (X,Y )) , for the finite field F of characteristic p generated by a square root of −c(D). Moreover, the trivial equality ϕ1((X − Y p)(Y −Xp)) = 0 ∈ F[X,X −1, Y, Y −1] (ΨΓ,p,tD (X,Y )) can be read as −c(D) 4Xp+1Y p+1 · (Y − (−c(D))(p−1)/2Xp) · (X − (−c(D))(p−1)/2Y p) · ((−c(D))(p−1)/2 − Y Xp) · ((−c(D))(p−1)/2 −XY p) = 0, 64 Kroneckerian polynomials and this one translates into a divisibility condition on F[X,Y ], and then on Z/(p)[X,Y ]: the polynomial ΨΓ,p,tD (X,Y ) divides the four-term product (Y − (−c(D))(p−1)/2Xp) · (X − (−c(D))(p−1)/2Y p) ·((−c(D))(p−1)/2 − Y Xp) · ((−c(D))(p−1)/2 −XY p). Now, remember that the polynomial ΨΓ,p,tD (X,Y ) must be symmetric and of degree at most p + 1 in each variable. Since all four factors of the product are irreducible and Z/(p)[X,Y ] is a unique factorization domain, this leaves us with only the two possibilities ΨΓ,p,tD (X,Y ) = (Y − (−c(D))(p−1)/2Xp) · (X − (−c(D))(p−1)/2Y p) or ΨΓ,p,tD (X,Y ) = ((−c(D))(p−1)/2 − Y Xp) · ((−c(D))(p−1)/2 −XY p), up to a non-zero multiplicative constant. Next, we take into account the expression uD,2 = ￿ c(D) t2D − 1 2tD and, work- ing as above with the ring homomorphism ϕ˜2 : Z[X,Y ] −→ Z[ ￿ c(D), 12 ][X,X −1, Y, Y −1] defined by ϕ˜2(X) := ￿ c(D) X2 − 1 2X , ϕ˜2(Y ) := ￿ c(D) Y 2 − 1 2Y , we reach a similar condition; namely, that the same polynomial ΨΓ,p,tD (X,Y ) divides the four-term product (Y − c(D)(p−1)/2Xp) · (X − c(D)(p−1)/2Y p) ·(c(D)(p−1)/2 + Y Xp) · (c(D)(p−1)/2 +XY p) in Z/(p)[X,Y ], and this leaves us only with the two possibilities ΨΓ,p,tD (X,Y ) = (Y − c(D)(p−1)/2Xp) · (X − c(D)(p−1)/2Y p) or ΨΓ,p,tD (X,Y ) = (c(D) (p−1)/2 + Y Xp) · (c(D)(p−1)/2 +XY p), up to a non-zero multiplicative constant. Finally, dealing with the possible values of ￿ −1 p ￿ and ￿ c(D) p ￿ and taking into account that ￿ −1 p ￿ = (−1)(p−1)/2 and ￿ c(D) p ￿ ≡ c(D)(p−1)/2 (mod p), we obtain the desired result. Chapter 3 The curves X(D, 1) of genus 0 In this chapter we study in detail the curves X(D, 1) of genus 0, which are those with D = 6, 10, 22, including their Atkin-Lehner quotients. First of all, we focus on the problem of computing fundamental domains for the groups ￿Γ(D,N),W ￿ ⊂ PSL(2,R), for W ⊂ WD,N . There are different approaches to effectively compute fundamental domains for them, for example, [Als99], [Joh00], [Voi09]. We are going to give an algorithm to compute them based on Ford’s method (cf. [For29]). We will use it to explicitly compute fundamental domains for the groups ￿Γ(D, 1),WD￿, D = 6, 10, 22. Next, we introduce q-expansions of automorphic functions for Γ(D,N) with respect to some suitable parameters q around CM points in H following the ideas in [Elk98], [BT07b]. In particular, this approach has been partially used to explicitly uniformize the curvesX(D,N)WDN of genus 0 such that there exists N ￿ > 1 with X(D,NN ￿)WDN of genus 0, cf. [Yan13], [Tu14]. In the following section we study how it can be done in the cases D = 6, 10, N = 1 and use the ideas in [SD77] together with the properties of the kroneckerian polynomials introduced in the previous section to uniformize X(22, 1)W22 , for which there is no N > 1 with g(X(22, N)W22) = 0. Even though the complexity of the computations increases dramatically whenever the number of elliptic cycles of the curve X(D,N)WDN increases, this approach does not rely on the existence of a suitable cover of genus 0, and therefore it can be used, at least, to uniformize Shimura curves X(D,N)WDN of genus 0 with a small number of elliptic cycles. For more details see Chapter 4. 65 66 The curves X(D, 1) of genus 0 Finally we review how the Hauptmoduln constructed relate to Igusa invari- ants in the cases D = 6, 10, cf. [BT08], [BG08], and extend the results to the case D = 22. In order to do so, we have to extend some results in [HM95] to this case using the Humbert surfaces computed in [Gru10]. 3.1 Fundamental domains for Γ(D,N) Let H = ￿ α,β Q ￿ , α > 0, be a rational indefinite quaternion algebra of discrimi- nant D and fix the embedding Φ : H ￿→M(2,R) given by Φ(x+ yi+ zj + tij) = ￿ x+ y √ α z + t √ α β(z − t√α) x− y√α ￿ . Let also N be a natural number such that gcd(D,N) = 1. Let O(D,N) be an Eichler order in H of level N . Let Γ = Γ(D,N) ⊂ PSL(2,R) be the correspond- ing fuchsian group. We will briefly deal with the computational part of the determination of fundamental domains for these groups as introduced in Section 1.2.3. Even though fundamental domains could be mostly avoided in the computations we will need later on, they provide a useful way to work with the set of complex points of the Shimura curves from an analytic viewpoint, for example, when dealing with numerical approximations. Since the hyperbolic volume of a fundamental domain is a known invariant of the group (Theorem 1.10) and the hyperbolic volume of a hyperbolic poly- gon can be easily computed by means of the Lambert formula (Theorem 1.8), Theorem 1.9 allows us to effectively compute fundamental domains for these fuchsian groups. However, since Theorem 1.9 involves the intersection of an infinite number of sets from which only a finite number are relevant, it is convenient to study how to order the sets involved. Fix t > 0 such that ti is not elliptic by Γ and consider γ0 = ￿ 1 0 1 ti −1 ￿ . Then, we are interested in computing γ0( ￿ γ∈Γ Eγ0γγ0) ∩H = H ￿ γ∈Γ γ0(Eγ0γγ0), 3.1 Fundamental domains for Γ(D,N) 67 where Eγ = (C￿ Cγ) ∪ {∞} and Cγ = {z ∈ C : |cz + d| ≤ 1}, where ￿ a b c b ￿ , with ad− bc = 1, is a representative for γ. Lemma 3.1. Assume that γ can be represented by the matrix ￿ a b c d ￿ , with ad− bc = 1. Define O, r ∈ R, r > 0, by O =− ab+ cdt 2 a2 + c2t2 − 1 , r2 =− b 2 + (d2 − 1)t2 a2 + c2t2 − 1 +O 2 = c2t4 + (a2 + d2 − 2)t2 + b2 (a2 + c2t2 − 1)2 . Then, H ∩ γ0(Eγ0γγ0) = ￿ {z ∈ H : |z −O| ≥ r}, if a2 + ct2 − 1 > 0, {z ∈ H : |z −O| ≤ r}, if a2 + ct2 − 1 < 0. Proof. If we write γ0 ￿ a b c d ￿ γ0 = ￿ A B C D ￿ , then γ0(Eγ0γγ0) = {z ∈ C : |Cγ0(z) +D| ≤ 1} and the result follows from a straightforward computation. The second expres- sion for r2 also follows from a direct computation if we take into account that ad− bc = 1. Once we have found some reasonable expressions for the sets appearing in the intersection, we are interested in choosing a suitable order to intersect the sets. Since the point ti is contained by construction in the interior of the fundamental domain, we will make the intersection according to the hyperbolic distance dH from the point ti to the boundary of the circles. Lemma 3.2. Let O, r ∈ R, r > 0, and denote by CircO(r) the circumference of center O and radius r. Then, dH(ti,CircO(r)) = arcsinh ￿ |O2 + t2 − r2| 2rt ￿ . 68 The curves X(D, 1) of genus 0 Proof. As we recalled in Section 1.2.1, the hyperbolic circumference of center ti and radius R coincides with the euclidean circumference of center cosh(R)ti and radius sinh(R)t. Thus, the hyperbolic distance of ti to CircO(r) is the only value of R for which these two circumferences are tangent. The two circumferences are tangent if and only if d(O, cosh(R)ti) = r ± sinh(R)t, which squaring both sides yields O2 + cosh2(R)t2 = r2 + sinh2(R)t2 ± 2r sinh(R)t and the desired formula is obtained. Finally, given a certain distance R > 0, we need to compute all the trans- formations yielding isometric circles whose boundary is within distance R from ti, which according to the previous lemma is equivalent to finding all transfor- mations such that |O2 + t2 − r2| ≤ 2rt sinh(R). Using the expressions above for O and r in terms of a, b, c, d, we obtain that |O2 + t2 − r2| = ￿￿￿￿b2 + (d2 − 1)t2a2 + c2t2 − 1 + t2 ￿￿￿￿ = |a2 + c2t2 − 1|r2. Therefore |O2 + t2 − r2| ≤ 2rt sinh(R) if and only if |a2 + c2t2 − 1|r2 ≤ 2rt sinh(R), which is equivalent to (|a2 + c2t2 − 1|r)2 = c2t4 + (a2 + d2 − 2)t2 + b2 ≤ 4t2 sinh2(R). In particular, we obtain that a2 + d2 − 2 ≤ 4 sinh2(R). Since a and d are conjugated quadratic with bounded denominators, this yields a finite number of possible choices for a, d. For any of these values the inequality above provides a bound for c2t4 + b2 = ( cβ ) 2(βt2)2 + b2 ≥ min(1,β2t4)(( cβ )2 + b2). Since cβ and b are conjugated quadratic as before, we obtain as well a finite number of choices for b, c. Consequently, we obtain the following result. Proposition 3.1. Fix Γ = Γ(D,N), t and γ0 as before. Then, for a given R > 0, the transformations γ such that the hyperbolic distance from ti to the boundary of γ0(Eγ0γγ0) is less or equal than R can be explicitly computed. 3.1 Fundamental domains for Γ(D,N) 69 Thus, after fixing R, all this procedure can be carried out and, if the volume of the region enclosing ti is equal to the volume of the fundamental domain, we have obtained indeed a fundamental domain for the group. Otherwise, R has to be increased, the new circles have to be added and the new volume has to be computed. Since the fundamental domain is limited by a finite number of circumferences for a big enough R the process ends. It is also worth noting that when we deal with the cases D > 1, the funda- mental domain will be a compact polygon in H. Thus, as soon as we have a compact polygon in H, we could prove that it is indeed a fundamental domain, without using the volume, choosing R such that this polygon is contained in the hyperbolic disc of center ti and radius R. Then, if we check that for all transfor- mations giving isometric circles within distance R from ti they do not further reduce the polygon, we can conclude as well that it is indeed a fundamental domain. This procedure can be easily modified to make it work for groups obtained attaching Atkin-Lehner involutions to the groups Γ(D,N). We will not write it down explicitly because we will not make use of it. Essentially, for every invo- lution of level m in the group, we would have to additionally consider matrices as above with all entries divided by √ m. Note that even though the entries of the matrices would be no longer quadratic all the assertions still hold. Next, we will give explicitly a fundamental domain for Γ(22, 1) which we have computed using this procedure. The details of the presentation of the quaternion algebra and the Eichler order chosen are given in the next section, together with some other cases. Here we will only illustrate the procedure above and show the resulting region. First of all, we choose t = 12 , sinh(R) = 6 and then: • the number of circles within distance R is 36; • the number of circles bordering the fundamental domain is 12. With these values, the figure we obtain when we print all 36 circles is the following: 70 The curves X(D, 1) of genus 0 Figure 3.1: All circles within distance R and when we only consider the relevant circles we obtain a fundamental domain for Γ(22, 1): Figure 3.2: The circles in the border of the fundamental domain It can be checked that the volume of this polygon coincides with the volume of a fundamental domain. Notice that there exist two tiny sides, in an euclidean sense, on the lower central part of the polygon. 3.2 Some explicit fundamental domains in the cases D = 6, 10, 22 71 3.2 Some explicit fundamental domains in the cases D = 6, 10, 22 We will now give fundamental domains for some of the groups ￿Γ(D, 1),W ￿, where W ⊂WD, which will be used later on. Fundamental domains for Γ(6, 1) and Γ(10, 1) can be found in [AB04]. We will not reproduce them below and we will give instead fundamental domains for ￿Γ(6, 1),W6￿ and ￿Γ(10, 1),W10￿. These can easily be deduced from the former ones studying the action ofWD on them. We will begin by giving a fundamental domain for Γ(22, 1) and deducing from it a fundamental domain for ￿Γ(22, 1),W22￿. We could directly use the method above to compute a fundamental domain for the group ￿Γ(D, 1),W ￿, however it is faster and more convenient, in order to study the morphisms between the corresponding quotient curves, to compute them this way. Next we will state the results for the cases D = 6 and D = 10. As we have seen before, in order to compute fundamental domains we need to have an explicit description of the group Γ, and this means an explicit choice of the quaternion algebra H together with an order O⊂H and an explicit em- bedding Φ : O ￿→ M(2,R). Recall that if we have H = ￿ a,b Q ￿ , and we write {1, i, j, ij} the corresponding Q-basis with i2 = a, j2 = b, ij = −ji, then Φ : H → M(2,R) x+ yi+ zj + tij ￿→ ￿ x+ y √ a z + t √ a b(z − t√a) x− y√a ￿ provides an embedding. In the cases we are interested in, we can choose the fol- lowing particular representative of the isomorphism class of quaternion algebras of discriminant D D HD 6 ￿ 3,−1 Q ￿ 10 ￿ 2,5 Q ￿ 22 ￿ 11,−1 Q ￿ and in these quaternion algebras we can choose a maximal order O(D, 1) with 72 The curves X(D, 1) of genus 0 Z-basis {1, I, J,K} as follows: D I J K 6 i j (1 + i+ j + ij)/2 10 i (1 + j)/2 (i+ ij)/2 22 i j (1 + i+ j + ij)/2 Then, the group Γ = Γ(D, 1) is the image in PSL(2,R) of Φ(O(D, 1)1). We will denote by (a, b, c, d)D the element a + bI + cJ + dK and [a, b, c, d]D the image of Φ(a + bI + cJ + dK) in PSL(2,R). We will drop the subscripts when no confusion arises. Finally, we need to give an explicit description of the elements in the group WD = {ωk}k|D, where ω1 = id and ωk is the transformation attached to a certain normalizing element of norm k in O(D, 1). The action induced on Γ(D, 1)\H is involutive and independent of the chosen element. We choose the following representatives: D ω2 ωD/2 ωD = ωD/2ω2 6 [1, 0, 1, 0]6 [2, 1, 2,−1]6 [0, 1, 3, 0]6 10 [0, 0, 0, 1]10 [−1, 1, 2,−2]10 [4,−2, 2,−1]10 22 [1, 0, 1, 0]22 [4,−3,−7, 3]22 [11,−3, 0, 0]22 Proposition 3.2. Let D(22, 1) be the hyperbolic decagon with vertices (Q22,i)10i=1 which, if no confusion is possible, we will simply denote by (Qi)i: Q22,1 = i, Q22,2 = 1 10 ￿ i− 3√11￿ , Q22,3 = i√3−5√1117+√11 , Q22,4 = i √ 3−√11 5+ √ 11 , Q22,5 = i √ 3−√11 17+5 √ 11 , Q22,6 = i ￿ 10− 3√11￿ , Q22,6+j = −Q22,6−j , j = 1, . . . , 4. (1) The internal angles at its vertices are, respectively,￿ π, π4 , π 3 , 2π 3 , π 3 , π 2 , π 3 , 2π 3 , π 3 , π 4 ￿ . (2) All its vertices are elliptic points, Q1, Q2, Q6, Q10 of order 2 and Q3, Q4, Q5, Q7, Q8, Q9 of order 3. The corresponding transformations of Γ(22, 1) fixing 3.2 Some explicit fundamental domains in the cases D = 6, 10, 22 73 Q22,1 Q22,2 Q22,3 Q22,4 Q22,5 Q22,6 Q22,7 Q22,8 Q22,9 Q22,10 db ec ec db a a Figure 3.3: Γ(22, 1)\H them are: γ22,1 = [0, 0, 1, 0]22, γ22,2 = [0,−3,−10, 0]22, γ22,3 = [0, 3, 9,−1]22, γ22,4 = [0,−1,−3, 1]22, γ22,5 = [2, 3, 11,−5]22, γ22,6 = [3, 3, 13,−6]22, γ22,7 = [2, 2, 11,−5]22, γ22,8 = [0, 0,−3, 1]22, γ22,9 = [0,−2, 9,−1]22, γ22,10 = [0, 3,−10, 0]22. (3) There are 6 elliptic cycles, two of order 2: {Q1}, {Q2, Q6, Q10}; and four of order 3: {Q3, Q5}, {Q4}, {Q7, Q9}, {Q8}. (4) The identifications between edges are given by: (Q1Q2, Q1Q10) by means of γ22,1, (Q2Q3, Q6Q5) by means of γ22,5 · γ22,4 = [3,−3,−7, 3]22, (Q3Q4, Q5Q4) by means of γ22,4, (Q6Q7, Q10Q9) by means of γ22,8 · γ22,7 = [3, 0,−7, 3]22, (Q7Q8, Q9Q8) by means of γ22,8. 74 The curves X(D, 1) of genus 0 (5) The transformations {γ22,1, γ22,5, γ22,4, γ22,7, γ22,8} together with the rela- tions 1 = γ222,1 = γ 3 22,4 = γ 3 22,8 = (γ22,8 · γ22,7 · γ22,5 · γ22,4 · γ22,1)2 = γ322,5 = γ322,7 give a presentation for the group Γ(22, 1). Remark. This fundamental domain was computed after the polygon found in the previous section. Some parts of the polygon were moved around in order to eliminate accidental cycles and to obtain a more workable polygon, without those tiny sides (in an euclidean sense). However, once the above set is given, all the listed properties, which are easily checked, already ensure that it is indeed a fundamental domain for Γ(22, 1). In order to deduce a fundamental domain for ￿Γ(22, 1),W22￿, we study the action of the group W22 on the domain D(22, 1). Using the explicit choices of ωk we have made above, we obtain: ω2(Q1, Q2, Q3, Q4, Q5, Q6) = (Q1, Q6, Q7, Q8, Q9, Q10), ω11(Q1, Q2, Q3, Q4) = (Q6, Q1, Q4, Q5), ω11γ22,1(Q1, Q8, Q9, Q10) = (Q6, Q7, Q8, Q1). It is now clear that (Q1, Q4, Q5, Q6) constitutes a fundamental domain for ￿Γ(22, 1),W22￿. Proposition 3.3. Let D(22, 1)+ be the hyperbolic hexagon with vertices (Q+22,i)6i=1 which, if no confusion is possible, we will simply denote by (Q+i )i: Q+22,1 = Q22,1, Q + 22,2 = Q22,4, Q + 22,3 = −1+i 3+ √ 11 , Q+22,4 = Q22,5, Q + 22,5 = Q22,6, Q + 22,6 = i √ 2 3+ √ 11 . (1) The internal angles at its vertices are, respectively,￿ π 4 , π 3 ,π, π 3 , π 4 ,π ￿ . (2) All its vertices are elliptic points: Q+3 , Q + 6 of order 2, Q + 2 , Q + 4 of order 3 and Q+1 , Q + 5 of order 4. The corresponding transformations of ￿Γ(22, 1),W22￿ fixing them are: γ+22,1 = ω2, γ + 22,2 = γ22,4, γ+22,3 = ω11 · γ222,4, γ+22,4 = γ22,5, γ+22,5 = ω2 · [−7,−3,−6, 3]22, γ+22,6 = ω22 · γ22,1. 3.2 Some explicit fundamental domains in the cases D = 6, 10, 22 75 Q+22,1 Q+22,2 Q+22,4 Q + 22,5 Q+22,3 Q+22,6 c c a b ab Figure 3.4: ￿Γ(22, 1),W22￿\H (3) There are 4 elliptic cycles, two of order 2: {Q+3 }, {Q+6 }; one of order 3: {Q+2 , Q+4 }; and one of order 4: {Q+1 , Q+5 }. Q+3 , Q+6 are CM points by the orders Z[ 1+ √−11 2 ] and Z[ √−22] respectively. (4) The identifications between edges are given by: (Q+1 Q + 2 , Q + 5 Q + 4 ) by means of ω11, (Q+2 Q + 3 , Q + 4 Q + 3 ) by means of γ + 22,3, (Q+5 Q + 6 , Q + 1 Q + 6 ) by means of γ + 22,6. (5) The transformations in (4) together with the relations 1 = (γ+22,3) 2 = (γ+22,6) 2 = (γ+22,3 · ω11)3 = (γ+22,6 · ω11)4 give a presentation for the group ￿Γ(22, 1),W22￿. (6) Any element of norm −1 in O(22, 1) induces a symmetry of the fundamental domain, which is the inversion with respect to the circumference with center at the origin and which passes through Q+3 , Q + 6 . 76 The curves X(D, 1) of genus 0 We are going now to reproduce the result for the cases D = 6, 10. The case D = 6 can be found in full detail in [BT07b], but it will be useful for future reference and to fix the notations to have all these cases together. Proposition 3.4. Let D(6, 1)+ be the hyperbolic quadrilateral with vertices (Q+6,i) 4 i=1 which, if no confusion is possible, we will simply denote by (Q + i )i: Q+6,1 = i, Q + 6,2 = −1+i 1+ √ 3 , Q+6,3 = (2− √ 3)i, Q+6,4 = √ 2i 1+ √ 3 . Q+6,1 Q+6,2 Q+6,3 Q+6,4 b b a a Figure 3.5: ￿Γ(6, 1),W6￿\H (1) The internal angles at its vertices are, respectively,￿π 4 , π 3 , π 4 ,π ￿ . (2) All its vertices are elliptic points: Q+4 of order 2, Q + 1 , Q + 3 of order 4 and Q+2 of order 3. The corresponding transformations of ￿Γ(6, 1),W6￿ fixing 3.2 Some explicit fundamental domains in the cases D = 6, 10, 22 77 them are: γ+6,1 = ω2, γ + 6,2 = ω3, γ+6,3 = ω2 · [2, 1, 1,−1]6, γ+6,4 = ω2 · ω3 = ω6 · [2, 1, 1,−1]6. (3) There are 3 elliptic cycles, one of order 2, {Q+4 }, one of order 4, {Q+1 , Q+3 }, and one of order 6, {Q+2 }. Q+4 is a CM point by Z[ √−6]. (4) The identifications between edges are given by: (Q+1 Q + 2 , Q + 3 Q + 2 ) by means of γ + 6,2, (Q+3 Q + 4 , Q + 1 Q + 4 ) by means of γ + 6,4. (5) The transformations in (4) together with the relations 1 = (γ+6,4) 2 = (γ+6,4 · γ+6,2)4 = (γ+6,2)6 give a presentation for the group ￿Γ(6, 1),W6￿. (6) Any element of norm −1 in O(6, 1) induces a symmetry of the fundamental domain, which is the inversion with respect to the circumference with center at the origin and which passes through Q+2 , Q + 4 . Proposition 3.5. Let D(10, 1)+ be the hyperbolic hexagon with vertices (Q+10,i)6i=1 which, if no confusion is possible, we will simply denote by (Q+i )i: Q+10,1 = i(1+ √ 2)√ 5 , Q+10,2 = 2−i√6 5(−2+ √ 2) , Q+10,3 = − 15 + 2i5 , Q+10,4 = −2+i √ 6 5(2+ √ 2) , Q+10,5 = i(−1+ √ 2)√ 5 , Q+10,6 = i√ 5 . (1) The internal angles at its vertices are, respectively,￿ π 2 , π 3 ,π, π 3 , π 2 ,π ￿ . 78 The curves X(D, 1) of genus 0 Q+10,1 Q+10,2 Q+10,3 Q+10,4 Q+10,5 Q+10,6 a c c a b b Figure 3.6: ￿Γ(10, 1),W10￿\H (2) All its vertices are elliptic points: Q+1 , Q + 3 , Q + 5 , Q + 6 of order 2 and Q + 2 , Q + 4 of order 3. The corresponding transformations of ￿Γ(10, 1),W10￿ fixing them are: γ+10,1 = ω5 · [3,−2, 0, 0]10, γ+10,2 = [0, 0, 1, 1]10, γ+10,3 = ω2, γ + 10,4 = [1, 0,−1, 1]10, γ+10,5 = ω5, γ + 10,6 = ω10 · [1, 0,−1,−1]10. (3) There are 4 elliptic cycles, three of order 2: {Q+1 , Q+5 }, {Q+3 }, {Q+6 }; and one of order 3, {Q+2 , Q+4 }. Q+1 , Q+5 , Q+3 and Q+6 are CM points by Z[ √−5], Z[ √−5], Z[√−2] and Z[√−10], respectively. (4) The identifications between edges are given by: (Q+1 Q + 2 , Q + 5 Q + 4 ) by means of γ210,4 · ω2, (Q+2 Q + 3 , Q + 4 Q + 3 ) by means of γ + 10,3, (Q+5 Q + 6 , Q + 1 Q + 6 ) by means of γ + 10,6. (5) The transformations in (4) together with the relations 1 = (γ+10,6 · γ210,4 · ω2)2 = (γ+10,3)2 = (γ+10,6)2 = (γ+10,3 · γ210,4 · ω2)3 3.3 q-expansions of automorphic functions 79 give a presentation for the group ￿Γ(10, 1),W10￿. (6) Any element of norm −1 in O(10, 1) induces a symmetry of the fundamental domain, which is the inversion with respect to the circumference with center at the origin and which passes through Q+3 , Q + 6 . 3.3 q-expansions of automorphic functions Now, we are going to introduce the q-expansions of automorphic functions. First of all, we define the local parameters around a given point P ∈ H, which we will use to obtain expansions for automorphic functions. Definition ([BT07b]). For P ∈ H and κ ∈ C∗, we define the local parameter around P with local constant κ, q = qP,κ : H→ C, as q(z) = κ z − P z − P . We will consider power series expansions of automorphic functions and forms with respect to these parameters, which we will suitably normalize later on. First of all, let us deal with the case of elliptic points. Proposition 3.6. Let P ∈ H, q = qP,κ and γ ∈ GL+(2,R) such that γ(P ) = P . Then, there exists ￿ ∈ C∗ such that q ◦ γ = ￿q. Moreover, ￿ = det γα2 , where α ∈ C is a root of the characteristic polynomial of γ. In particular, if P is a CM point by a quadratic imaginary field K, and we consider an embedding φ : K ￿→ {δ ∈ SL(2,R) : δ(P ) = P}, then α = φ−1(γ). Proof. The existence of ￿ together with an expression for it are a simple com- putation and the rest of the properties can be checked immediately from it. Observe, in particular, that if Γ is a fuchsian group and P is an elliptic point of order e by Γ, then an automorphic function with respect to Γ has an expansion as a power series in qe. We will compute expansions of automorphic functions as solutions of a cer- tain type of differential equations. Let us now introduce the automorphic deriva- tive, as introduced in [BT07a], which is closely related to the classical schwarzian derivative. 80 The curves X(D, 1) of genus 0 Definition. The automorphic derivative of a meromorphic function f is Da(f, z) = 2f ￿f ￿￿￿ − 3(f ￿￿)2 (f ￿)4 , where f ￿ = dfdz is the usual derivative. Some basic properties of this differential operator are the following. Proposition 3.7. (1) If g is automorphic with respect to Γ, the same is true for Da(g, z). (2) If g and h are meromorphic functions and its composition is defined, Da(h ◦ g, z) = Da(h, g(z)) + Da(g, z) (g￿)2 . (3) If γ is an homographic transformation, Da(γ(z), z) = 0. (4) Let X, Y be meromorphic functions such that there exists a polynomial P in two indeterminates which satisfies P (X,Y ) = 0. Then, it holds that Da(X, z) = P 2X P 2Y Da(Y, z) + Θ(X,Y ) P 2XP 4 Y , where the subscripts of P denote partial derivatives and Θ(X,Y ) =6P 2XP 3 Y PXXY + 3P 2 XXP 4 Y − 2PXP 3Y (3PXXPXY + PXXXPY ) − 6P 3XPY (PXY Y PY − PXY PY Y )− P 4X(3P 2Y Y − 2PY PY Y Y ). Observe, in particular, that given a function f and a local parameter qP around a point P , Da(f, z) = Da(f, qP (z)). The following result is a reformulation of the classical result concerning the conformal mapping of hyperbolic polygons which can be found, for example, in [Neh75]. When these polygons are obtained from suitable cocompact fuchsian groups, see [BT07b] for a more clear approach along this lines. A discussion in the lines of our formulation can be found in [Elk98]. 3.3 q-expansions of automorphic functions 81 Proposition 3.8. Let Γ be a fuchsian group with elliptic cycles c1, . . . , cr of orders e1, . . . , er. Assume that Γ\H is compact and has genus 0 and that f is a generator of the function field A0(Γ)/C. Then, Da(f, z) +R(f) = 0 for R(f) ∈ C(f) of the form R(f) = ￿ i 1− 1/e2i (f − ai)2 + ￿ i Bi f − ai , where the sums run over the elliptic cycles where f takes finite values ai. More- over, if the values at all the elliptic cycles are finite, the following conditions on Bi hold: (1) ￿ iBi = 0, (2) ￿ i aiBi + ￿ i(1− 1/e2i ) = 0, (3) ￿ i a 2 iBi + ￿ i ai(1− 1/e2i ) = 0. On the other hand, if f is infinite at an elliptic cycle of order e, then these conditions must be replaced by: (1) ￿ iBi = 0, (2) ￿ i aiBi + ￿ i(1− 1/e2i )− (1− 1/e2) = 0. Proof. Let Pi ∈ H, i = 1, . . . , r, be a point representing the elliptic cycle ci and let qi = κi z−Piz−P i , κi ∈ C, be a local parameter around the point Pi. The value of κi will be fixed later. Since Da(f(z), z) is a meromorphic automorphic function with respect to Γ and f is a generator of the field of automorphic functions with respect to Γ, Da(f(z), z) ∈ C(f(z)). It is immediate to check that the poles of f are zeros of Da(f(z), z), therefore the set of singularities of the function Da(f(z), z) = 2f ￿(z)f ￿￿￿(z)− 3(f ￿￿(z))2 (f ￿(z))4 is contained in the set of double points of f which are not poles, and this is nothing else than the set of elliptic points of Γ where f takes finite values. 82 The curves X(D, 1) of genus 0 Fix Pi and assume that f(Pi) = ai ∈ C. There exists κi ∈ C such that f = ai+q ei i +ai,2q 2ei i +. . . and then an expansion forDa(f(z), z) = Da(f(qi), qi) around Pi has leading term −(1−1/e2i )q−2eii equal to that of −(1−1/e2i ) 1(f−ai)2 . Therefore, there exists Bi such that Da(f(z), z) + 1− 1/e2i (f − ai)2 + Bi f − ai has the same singularities as Da(f(z), z) except for Pi. Thus, repeating this process for all the points Pi where f takes a finite value ai, we obtain that Da(f(z), z) + ￿ i 1− 1/e2i (f − ai)2 + ￿ i Bi f − ai is holomorphic and automorphic, and therefore constant. That is, there exists γ ∈ C such that Da(f(z), z) + ￿ i 1− 1/e2i (f − ai)2 + ￿ i Bi f − ai + γ = 0. (∗) Let us now study this equality around a point P ∈ H where f has a pole. We will make a distinction on whether P is an elliptic point for Γ or not. We will give the proof in the case P non-elliptic. The case P elliptic is totally analogous. Consider an expansion f = 1q + ￿ k ckq k. In this situation, Da(f(z), z) has a zero of order at least 4 at P . If we compute the corresponding expansion of (∗), we obtain 0 =Da(f(z), z) + ￿ i 1− 1/e2i (f − ai)2 + ￿ i Bi f − ai + γ =γ + ￿￿ i Bi ￿ q + ￿￿ i (1− 1/e2i ) + ￿ i aiBi − c0 ￿ i Bi ￿ q2 + ￿ 2 ￿ i (1− 1/e2i )ai + ￿ i a2iBi − 2c0( ￿ i (1− 1/e2i ) + ￿ i aiBi) ￿ q3 + . . . from which γ = 0 and the equations in the statement of the proposition follow. 3.3 q-expansions of automorphic functions 83 Remark. Observe that, if there are r elliptic cycles, then R depends on 2r parameters (or 2r − 1 if f is not finite at an elliptic cycle). The Hauptmodul f is determined by its value at 3 different points of Γ\H, which we can choose to be 3 of the elliptic cycles, and then the conditions on the Bi leaves us with 2(r− 3) unknown parameters. In conclusion, there are no unknown parameters after choosing f properly, if and only if Γ has only 3 elliptic cycles. This situation is exploited in [Elk98] and [BT07b] to deal with the case Γ = ￿Γ(6, 1),W6￿. However, neither the case D = 10 nor D = 22 fall into this particular situation, since in both cases we end up with 4 elliptic cycles. The exact computation of the parameters in this differential equation, classi- cally called accessory parameters, is a problem to which no general solution is known (cf. [FK12, SD77]). We will see later on how we can exploit the algebraic structures present in the curves we are interested in to explicitly compute these unknown values. It is worth observing that for a Hauptmodul f defined over Q of any of the curves X(D, 1)W , D = 6, 10, 22, {id} ￿= W ⊂ WD, and Γ the corresponding fuchsian group, the set of points of Γ\H where f takes real values, which corre- sponds to the set of real points of the curve, can be identified with the set of fixed points in Γ\H of the transformation attached to any element in O(D, 1)−1. This set of fixed points, when the fundamental domain is properly chosen, yields the sides of an hyperbolic polygon, hence providing the link between the statement given above, and the classical result on conformal mapping. Another crucial observation which we will use extensively later on is the following. Given f an automorphic function for Γ and P ∈ H an elliptic point of order e ≥ 2, then there exists a local parameter q = qP,κ for a certain κ ∈ C∗ such that f(q) = a0+qe+ ￿ k≥2 akq ke. In particular, if we know the correspond- ing value a0 of the function f at P , and we know the automorphic derivative of f , we can explicitly compute the expansion of f up to a desired order with respect to this parameter q. We will see later on that the determination of the value a0 can be done exactly, however we can only compute in general the value of κ through approximations. In the case D = 6 the values of κ for all the elliptic points can be found in [BT08]: it is a trascendental value, whose trascendency class can be obtained by means of the Chowla-Selberg formula (cf. [SC67]), and its exact value is a special value of a suitable hypergeometric series. However, this approach using hypergeometric series can not be extended to the cases D = 10, 22. In Chapter 6 we will see as well how to compute expan- sions around arbitrary CM points, for suitable local parameters q, starting from 84 The curves X(D, 1) of genus 0 the information around the elliptic points and making use of the kroneckerian polynomials. The exact value of the constants of the local parameters, as in the case of the elliptic points, can be approximated up to a desired precision, but no approach to obtain the exact value is known. 3.3.1 Solutions of automorphic differential equations by means of order 2 linear differential equations Series expansions around a given CM point for the Hautpmoduln we introduced before can be computed directly from the automorphic differential equation once the initial conditions are known. However, the recursion formula is rather involved and for some computations it will be useful to obtain these series expan- sions as solutions of simpler differential equations. These are classical results which go back to Schwarz and whose relation to the theory of automorphic functions was already well-known at the end of the 19th century. Proposition 3.9 ([FK12, For02]). Let R(t) ∈ C(t) be a rational function and f : H→ C∪ {∞} a meromorphic function. Consider a branch of the functional inverse of f , f−1 : C ∪ {∞}→ H, and denote w = f(z). Then, f is a solution of the automorphic differential equation Da(f(z), z)+R(f(z)) = 0 if and only if f−1(w) = u1(w)u2(w) , where u1, u2 are solutions of the second order linear differential equation u￿￿ + R(w)4 u = 0. Let f be a Hauptmodul for a certain fuchsian group and assume that it is a solution of the automorphic differential equation Da(f(z), z)+R(f(z)) = 0, for a certain rational function R(t). Let P ∈ H be a CM point and denote by e its order (e = 1 if the point is non-elliptic). For simplicity, assume f(P ) = a ∈ C and consider a local parameter q = qP such that f = a+ qe + bq2e + . . . By the invariance of Da with respect to homographic transformations, we have that Da(f(q), q) +R(f(q)) = 0. Now we need to distinguish between the case e = 1 and e > 1: - e = 1: In this case, we can choose a couple of solutions u1, u2 such that u1(a) = u￿2(a) = 0, u￿1(a) = u2(a) = 1, and then f−1 = u1/(bu1 + u2). There- fore, if we compute expansions of u1, u2 in C[[w− a]], we obtain an expansion for f with respect to q by finding the inverse of the series corresponding to u1/(bu1 + u2). - e > 1: In this case, the linear differential equation has a singularity at the point w = a. However, since the leading term of the rational function R(w)/4 3.4 Hauptmoduln: t+D 85 around w = a is 1−1/e 2 4(w−a)2 , there exist solutions of this differential equation in (w − a)(e−1)/(2e)C[[(w − a)1/e]] such that u1 = (w − a)(e−1)/(2e)((w − a)1/e + ￿ k≥2 a1,k(w − a)k/e) and u2 = (w − a)(e−1)/(2e)(1 + ￿ k≥2 a2,k(w − a)k/e). Then, f−1 = u1/u2 and computing the inverse series, we obtain an expansion for f with respect to qe. The case f(P ) = ∞ can be reduced to the above discussion by considering the function 1/f which satisfies a differential equation of the same type with a different R, which according to Proposition 3.7 is R(1/t)/t4. 3.4 Hauptmoduln: t+D 3.4.1 An overview of the case D = 6 The case D = 6 can be found in detail in [BT07b]. We present here the outline of the process in this case, which we will reproduce next for the cases D = 10, 22, where an additional step will be needed. Let X(6, 1)+ be the quotient curve of X(6, 1) by W6 and Γ(6, 1)+ the corresponding group ￿Γ(6, 1),W6￿. This group has three elliptic cycles corresponding to the CM points by Z[i] (order 4), Z[ 1+ √−3 2 ] (order 6) and Z[ √−6] (order 2). Recall, in this situation, that in the proof of Theorem 2.4, we already made a choice for a Hauptmodul t+6 which took values 1 at the elliptic cycle corresponding to Z[i],∞ at that corresponding to Z[ 1+ √−3 2 ] and 0 at that corresponding to Z[ √−6]. We fix this Hauptmodul from now on. Proposition 3.8 tells us that there exists R+6 (f) ∈ C(f) such that Da(t+6 , z) +R + 6 (t + 6 ) = 0 and R+6 (f) = 1− 116 (f − 1)2 + 1− 14 f2 + B1 f − 1 + B3 f where B1 +B3 = 0, B1 + (1− 116 ) + (1− 14 )− (1− 136 ) = 0 which means R+6 (f) = 108− 113f + 140f2 144(−1 + f)2f2 . 86 The curves X(D, 1) of genus 0 Then, following the construction given in the proof of Theorem 2.4, we obtain the canonical model of X(6, 1) and the remaining quotients. That is, we define functions u6,1, u6,2, u6,3 = u6,1/u6,2 such that u26,1 = −3 t + 6 t+6 −1 and u26,2 = 3t+6 −1 , which provide canonical models for the curves X(6, 1)￿ω3￿, X(6, 1)￿ω6￿ and X(6, 1)￿ω2￿, respectively. Then, an affine chart of the canon- ical model of X(6, 1) is given by (u6,1 : u6,2 : 1). Moreover, since we have computed the automorphic derivative of t+6 , the automorphic derivatives of u6,k follow immediately from Proposition 3.7. For instance, in order to compute the automorphic derivative of u6,1, we consider the polynomial P (X,Y ) = X2(Y − 1) + 3Y which satisfies that P (u6,1, t+6 ) = 0 and we obtain that Da(u6,1, z) = (2u6,1(t + 6 − 1))2 (u26,1 + 3) 2 (−R+6 (t+6 (z))) + 3 u26,1 . Now using that t+6 = u26,1 3+u26,1 , Da(u6,1, z) + −113 + 9u26,1 12 ￿ 3 + u26,1 ￿2 = 0. Similarly, Da(u6,2, z) + 96− 49u26,2 + 9u46,2 12u26,2 ￿ 3 + u26,2 ￿2 = 0, Da(u6,3, z) + −103 + 32u26,3 36 ￿ 1 + u26,3 ￿2 = 0, Da(t6, z) + 27 + 74t26 + 27t 4 6 36(−1 + t6)2t26(1 + t6)2 = 0. Therefore, q-expansions for all these functions can be computed around any point in H which corresponds to an elliptic one in X(6, 1)+(C). In Chapter 6, we will recall some of these expansions and explain how to compute expansions around arbitrary CM points. 3.4 Hauptmoduln: t+D 87 Remark. It is worth noting that the function t+6 we have introduced above is not exactly the one given under the same name in [BT07b], which in terms of our t+6 is t + 6 /(t + 6 − 1). On the other hand, our functions u6,1, u6,2, u6,3 = u6,1u6,2 and t6 do correspond exactly to the functions u3, u6, u2 and t6. 3.4.2 The cases D = 10 and D = 22 In the previous section we have briefly introduced the case D = 6, the only one of the three we are dealing with which corresponds to a triangle group, and we have computed the automorphic derivatives for a set of functions providing the canonical models of X(6, 1) and all its quotients. The key point has been the knowledge of the automorphic derivative of a Hauptmodul of the genus 0 curve X(6, 1)+, from which the automorphic derivatives of all the other functions involved are easily computed. The situation is exactly the same for the cases we are now going to deal with, with the important difference that none of the curves X(D, 1), D = 10, 22, nor their quotients is triangular; the corresponding groups are still generated by elliptic transformations, but the number of elliptic cycles is always greater than 3. Firstly, we are going to treat the case D = 22 to show how we resolve the computation of the unknown parameters, and next we are going to sketch the case D = 10, which can be computed following exactly the same procedure, but has already been computed by other means in [Elk98] using that the curve X(10, 3)W10 is a genus 0 curve which gives a covering of X(10, 1)+ of degree 4 and studying its ramification. In the case D = 22 this approach cannot be used straightforwardly since the curve X(22, N)W22 does not have genus 0 for any N . Proposition 3.10. Let l ￿= 11 be an odd prime and r a positive integer. Then, the genus of the curve X(22, lr)W22 is equal to￿ 1 24 ￿ 5lr−1(l + 1) + 3− 6 ￿−11l ￿− 6 ￿−22l ￿− 9 ￿−1l ￿￿ , if l = 3, r ≥ 2, 1 24 ￿ 5lr−1(l + 1)− 5− 6 ￿−11l ￿− 6 ￿−22l ￿− 8 ￿−3l ￿− 9 ￿−1l ￿￿ , otherwise. In particular, g(X(22, N)) ≥ 1 for any integer N > 1 coprime to 22. Proof. The genus of the curve X(22, lr) can be computed using the formula in Theorem 1.10 and we obtain that 2− 2g(X(22, lr)) = ￿ − 53 lr−1(l + 1) + 1 + ￿−1 l ￿ , if l = 3, r ≥ 2, − 53 lr−1(l + 1) + 113 + ￿−1 l ￿ + 83 ￿−3 l ￿ , otherwise. 88 The curves X(D, 1) of genus 0 Now, we want to apply the Riemann-Hurwitz formula and therefore we need to study the ramification points of the morphism X(22, lr)→ X(22, lr)W22 , which has degree 4, or equivalently the set of fixed points by the transformations in W22. According to Theorem 4.1, the set of fixed points by the transformations in W22 are the CM points corresponding to the following rings of integers: Fixed points by ω2 : CM(Z[i]), Fixed points by ω11 : CM(Z[ 1+ √−11 2 ]), Fixed points by ω22 : CM(Z[ √−22]). Finally, we have to compute how many CM points by these rings there are. For this purpose, we use the formula given in Theorem 1.7 and obtain that #CM(Z[i]) = 2 ￿ 1 + ￿−1 l ￿￿ , #CM(Z[ 1+ √−11 2 ]) = 2 ￿ 1 + ￿−11 l ￿￿ , #CM(Z[ √−22]) = 2 ￿1 + ￿−22l ￿￿ . Since the ramification index of any of these points is 2, Riemann-Hurwitz for- mula gives the desired expression for the genus of X(22, lr)W22 . Let us, for the moment, try to follow the same procedure we used in the case D = 6. Let X(22, 1)+ be the quotient curve of X(22, 1) by W22 and Γ(22, 1)+ be the corresponding group ￿Γ(22, 1),W22￿. This group has four elliptic cycles corresponding to the CM points by Z[i] (order 4), Z[ 1+ √−3 2 ] (order 3), Z[ 1+ √−11 2 ] (order 2) and Z[ √−22] (order 2). Recall, in this situation, that in the proof of Theorem 2.4, we already made a choice for a Hauptmodul t+22 which took values 1 at the elliptic cycle corresponding to Z[i], ∞ at that corresponding to Z[ 1+ √−11 2 ] and 0 at that corresponding to Z[ √−22]. Proposition 3.7 tells us that there exists R+22(f) ∈ C(f) such that Da(t+22(z), z) +R+22(t+22(z)) = 0 and R+22(f) = 1− 116 (f − 1)2 + 1− 14 f2 + 1− 19 (f − a)2 + B1 f − 1 + B3 f + B f − a where B1 +B3 +B = 0, B1 + aB + (1− 116 ) + (1− 14 ) + (1− 19 )− (1− 14 ) = 0. Therefore, B1 = 1144 (−263 − 144aB) and B3 = −B1 − B. Then, if we write b = −216 + 47a− 144aB + 144a2B, we obtain that R+22(f) = 108a2 + abf + ￿ 20− 263a+ 27a2 − b− ab￿ f2 + (7− 7a+ b)f3 + 108f4 144(f − 1)2f2(f − a)2 . 3.4 Hauptmoduln: t+D 89 Now we have to compute the values of a, b and, in order to do so, we will use the kroneckerian polynomials introduced in Chapter 2. Since the function t+22 takes rational values at 3 of the four elliptic cycles, it takes a rational value at the fourth one as well and therefore a must be rational. Looking at the fundamental domain, we can see moreover that a ∈ (0, 1). Let us fix p = 3 ￿ D and consider the kroneckerian polynomial of level 3 attached to the Hauptmodul t+22 of X(22, 1)+, Ψ(X,Y ) = ΨΓ(22,1)+,t+22,3(X,Y ) ∈ Q[X,Y ], which is symmetric and has degree 4 in each of the variables, that is Ψ(X,Y ) = ￿4 i,j=0 ci,jX iY j with ci,j = cj,i ∈ Q. In this case, among the four elliptic cycles, only those by Z[ 1+ √−3 2 ] and Z[ 1+ √−11 2 ] are fixed by a transformation in O3 ·W22. Thus, we could now use Proposition 2.5 to impose some conditions on the ci,j , that is (X−a)X2 | Ψ(X,X). We need to consider expansions around these two elliptic cycles, t+22(q3) = a+ q 3 3 + ￿ k≥2 d3,kq 3k 3 , t+22(q11) = q 2 11 + ￿ k≥2 d11,kq 2k 11 , where q3(z) = κ3 z−P3z−P3 , q11(z) = κ11 z−P11 z−P11 are suitable local parameters around the points P3, P11 ∈ H representing the corresponding elliptic cycles. Using the automorphic differential equation we have computed above, we can find the values d3,k, d11,k ∈ Q(a, b) up to a desired value of k. Moreover, expansions for a function t+22 ◦ ω3 ◦ γ, γ ∈ Γ(22, 1), around these two points can be obtained easily from the ones for t+22. Let P be one of these two points and q the corresponding local parameter. Since there exists an element ω ∈ O3 · W22 fixing P , then t+22 ◦ ω3 ◦ γ = t+22 ◦ ω and q ◦ ω/q ∈ C. After Proposition 3.6, no computation is necessary to obtain the value of this quotient. In this particular case, the transformations ω can be chosen such these quotients are −1 in the case of q3 and −5− √−11 6 in the case of q11. Finally, we consider the equations 0 =Ψ(t+22(q3), t + 22(−q3)) =Ψ(t+22(q11), t + 22( 1 6 ￿−5−√−11￿ q11)) which provide a system of linear homogeneous equations in the unknown coeffi- cients of Ψ with coefficients in Q(a, b) which has to be indeterminate. The values 90 The curves X(D, 1) of genus 0 of a, b for which this occurs can be obtained in a straightforward (but lengthy) way, computing some minors and the corresponding resultants. Because of their size, none of the polynomials and resultants involved in the process can be re- produced here. The only possible rational value of a ∈ (1,∞) is a = 27/16, giving only a possible value b = −5899/16 and this yields R+22(f) = 8748− 17697f + 18316f2 − 10624f3 + 3072f4 16(−1 + f)2f2(−27 + 16f)2 . We obtain as well the kroneckerian polynomial of level 3 attached to t+22, namely Ψ(X,Y ) =55130625− 260884800X + 494913024X2 − 440745984X3 + 157351936X4 − 260884800Y + 848782592XY − 1089896448X2Y + 802750464X3Y − 308281344X4Y + 494913024Y 2 − 1089896448XY 2 + 693043200X2Y 2 − 245366784X3Y 2 + 150994944X4Y 2 − 440745984Y 3 + 802750464XY 3 − 245366784X2Y 3 − 117440512X3Y 3 + 157351936Y 4 − 308281344XY 4 + 150994944X2Y 4. Now we can, as in the caseD = 6, recover the automorphic derivatives for the rest of the functions u22,1, u22,2, u22,3, t22 introduced in Theorem 2.4 obtaining a whole set of generators for the curve X(22, 1) and its quotients, cf. Theorem 3.1. The outline of the case D = 10 is completely analogous, except for the computations being fairly easier. LetX(10, 1)+ be the quotient curve ofX(10, 1) byW10. There are also four elliptic cycles, three of them of order 2 corresponding to Z[ √−2],Z[√−5] and Z[√−10] and one of order 3 and therefore attached to Z[ 1+ √−3 2 ]. In Theorem 2.4 we chose a Hauptmodul t + 10 such that it takes the value 1 at the elliptic cycle corresponding to Z[ √−2], ∞ at that corresponding to Z[ √−5] and 0 at that corresponding to Z[√−10]. Then, Da(t+10(z), z) +R + 10(t + 10(z)) for a certain R+10(f) ∈ C(f) which can be written as R+10(f) = 27a2 + abf + (5− 59a− b− ab)f2 + (−5 + 5a+ b)f3 + 27f4 36(−1 + f)2f2(−a+ f)2 . As in the previous case, a must be a rational number, and in this case, observing the fundamental domain, we obtain that a > 1. We fix p = 3 ￿ 10 and 3.4 Hauptmoduln: t+D 91 we consider the kroneckerian polynomial of level 3 attached to t+10, Ψ(X,Y ) = ΨΓ,3,t+10(X,Y ) = ￿ 0≤i,j≤4 ci,jX iY j , with ci,j = cj,i. In this case, three of the elliptic cycles are fixed by transforma- tions in O3 ·W10, namely those attached to Z[ √−2],Z[ 1+ √−3 2 ] and Z[ √−5]. Let P2, P3, P5 respectively be points representing these elliptic cycles and consider local parameters as above q2, q3, q5 such that t+10(q2) = 1 + q 2 2 + ￿ k≥2 d2,kq 2k 2 , t+10(q3) = a+ q 3 3 + ￿ k≥2 d3,kq 3k 3 , t+10(q5) = q −2 5 + ￿ k≥0 d5,kq 2k 5 . Hence, we obtain, as above, 0 =Ψ(t+10(q2), t + 10( −1−2√−2 3 q2)), 0 =Ψ(t+10(q3), t + 10(−q3)), 0 =Ψ(t+10(q5), t + 10( −2−√−5 3 q5)). Finally, when we impose that this system of linear equations is indeterminate, we obtain the values a = 27/2, b = −1061/2, from which R+10(f) = 2187− 3183f + 3067f2 − 208f3 + 12f4 4(−1 + f)2f2(−27 + 2f)2 and Ψ(X,Y ) =29160000− 19926000X + 6525225X2 − 1066410X3 + 83521X4 − 19926000Y − 19848850XY + 9014010X2Y − 21744X3Y − 69360X4Y + 6525225Y 2 + 9014010XY 2 + 2121246X2Y 2 − 271440X3Y 2 + 14400X4Y 2 − 1066410Y 3 − 21744XY 3 − 271440X2Y 3 + 3200X3Y 3 + 83521Y 4 − 69360XY 4 + 14400X2Y 4. Let us sum up all these results in the following theorem, together with some additional properties we have skipped in the discussion above. 92 The curves X(D, 1) of genus 0 Theorem 3.1. Let D = 6, 10, 22. (1) Let f be any of the functions previously introduced in Theorem 2.4 or any of the functions t+D considered above. Then, there exists a rational function R(X) ∈ Q[X] such that Da(f(z), z) +R(f(z)) = 0 and the precise value of R(X) is given in the following table: f D 6 10 t+D 108−113f+140f2 144(−1+f)2f2 2187−3183f+3067f2−208f3+12f4 4(−1+f)2f2(−27+2f)2 uD,1 −113+9f2 12(3+f2)2 − 10(303−22f 2+7f4) (2+f2)2(27+f2)2 uD,2 96−49f2+9f4 12f2(3+f2)2 2(−1075+614f2+29f4) (−5+f)2(5+f)2(2+f2)2 uD,3 −103+32f2 36(1+f2)2 − 2859+5522f2+2675f4 (1+f2)2(27+25f2)2 tD 27+74f2+27f4 36(−1+f)2f2(1+f)2 128(1+23f2+f4) (1−52f2+f4)2 f D 22 t+D 8748−17697f+18316f2−10624f3+3072f4 16(−1+f)2f2(−27+16f)2 uD,1 −80619−6743f2−193f4+3f6 4(11+f2)2(27+f2)2 uD,2 −2816+8800f2+61f4+3f6 4(−4+f)2(4+f)2(11+f2)2 uD,3 − 10167+14240f2+5888f44(1+f2)2(27+16f2)2 tD 11(33+508f2+3014f4+508f6+33f8) 4f2(11−86f2+11f4)2 (2) The values of the functions t+D at the CM points by the rings of integers of some quadratic imaginary fields of small discriminants are given in the following table. The symbol ∗ indicates that no CM points by the given field 3.5 Expression of t+D in terms of absolute Igusa invariants 93 exist in the curve. −3 −4 −8 −11 −19 −20 −24 −40 t+6 ∞ 1 ∗ ∗ 32111024 ∗ 0 2312125 t+10 27 2 ∗ 1 ∗ ∗ ∞ ∗ 0 t+22 27 16 1 ∗ ∞ ∗ 516 ∗ ∗ Remark. The values of the functions above at a certain CM point can be ex- actly computed using Proposition 2.5 once a suitable kroneckerian polynomial is computed. In Chapter 6 we will discuss in detail the computation of kroneck- erian polynomials and the determination of the values of a Hautpmodul at a given CM point. 3.5 Expression of t+D in terms of absolute Igusa invariants Let D = 6, 10, 22 and let t+D be the Hauptmoduln for ￿Γ(D, 1),WD￿ introduced in Section 2.4. In [BG08], the uniformization of the curves X(D, 1)+, D = 6, 10, is carried out embedding these curves into Igusa’s threefold (cf. [Rot04]) and obtaining a Hauptmodul for them in terms of absolute Igusa invariants. In order to do so, they make use of the parametric families of genus 2 curves whose jacobian has quaternionic multiplication by O(D, 1), D = 6, 10, computed by Hashimoto and Murabayashi in [HM95]. Then, all this information is used in [BT08] to write the Hauptmodul, which according to our notations would be t+6 /(t + 6 − 1), in terms of Igusa invariants and Thetanullwerte. Firstly we are going to introduce some classical results on the Siegel upper half-plane and Igusa’s threefold and the natural embeddings of certain Atkin- Lehner quotients of Shimura curves into Igusa’s threefold. Next we are going to reproduce briefly the case D = 6 and then we will apply the same approach to the case D = 10, where all the tools used when D = 6 are also available. Finally we will deal with this problem for D = 22. We will begin by computing a parametric family of genus 2 curves playing the role of the Hashimoto-Murabayashi family in our case, using some of the explicit equations 94 The curves X(D, 1) of genus 0 of Humbert surfaces computed in [Gru10], and then we will be able to follow the outline used in [BG08] to obtain a Hauptmodul for X(22, 1)+ in terms of Igusa invariants. Finally, a comparison between the values of this Hauptmodul and the one we have already computed at certain CM points will allow us to write t+22 in terms of Igusa invariants. In this way, given a point P ∈ Γ(D, 1)\H ∼= X(D, 1)(C), we will obtain an equation for a genus 2 curve in terms of t+D(P ) whose jacobian corresponds to the abelian surface with quaternionic multiplication represented by P . Moreover, some additional properties of these curves will be discussed, for example, which of them have additional automorphisms or have simultaneously quaternionic multiplication by two of the orders O(D, 1), D = 6, 10, 22, we are dealing with. 3.5.1 From a point on X(D, 1) to the corresponding abelian surface Given an indefinite quaternion algebra over Q of discriminant D > 1, HD, an embedding Φ : HD → M(2,R) and a maximal order O ⊂ HD, we have introduced the group Γ(D, 1) and considered the quotient space Γ(D, 1)\H, which according to Shimura, cf. [Shi67], can be identified with the set of com- plex points of an algebraic curve defined over Q, X(D, 1), and interpreted as a (coarse) moduli space of principally polarized abelian surfaces with quaternionic multiplication by O. In order to make this identification, we need an auxiliary element ρ ∈ O such that ρ2 + D = 0, which necessarily normalizes O. In this way, the application HD → HD α ￿→ α∗ = ρ−1αρ is a positive involution acting on O. Then, the points of X(D, 1)(C) are in one to one correspondence with the set of isomorphism classes of (A, ι,L), where (A,L) is a principally polarized abelian surface and ι : O → End(A) is an injective ring homomorphism such that the Rosati involution with respect to L coincides with ∗ on ι(O). Note that the isomorphism class of the algebraic curve X(D, 1) does not depend on ρ, but the moduli interpretation, in general, does. On the other hand, we can consider the moduli space of principally polarized abelian surfaces, A2, which is an affine scheme of dimension 3 and contains, via the Torelli embedding, the moduli space of genus 2 curves, M2, as a Zariski open and dense subset. Similarly to the case of Shimura curves and therefore 3.5 Expression of t+D in terms of absolute Igusa invariants 95 to the classical modular case, the set of complex points of A2 can be identified with Sp(4,Z)\H2. Generators for the field of Siegel modular functions and the graded ring of modular forms where given by Igusa, cf. [Igu60, Igu62, Igu64], both in terms of Thetanullwerte and in terms of invariants of binary sextic forms in the subspace M2. In this case, the construction of the complex tori attached to the abelian surfaces is carried out in the same way it is done in the modular case. Therefore, we can naturally construct a map X(D, 1) → A2 such that [(A, ι,L)] ￿→ [(A,L)]. This map, obtained when we drop the quaternionic struc- ture, has been studied by Rotger in [Rot04] where it is proved that it factors through an Atkin-Lehner quotient of X(D, 1) giving rise to a birrational map. In all the cases D = 6, 10, 22, this quotient is the full Atkin-Lehner quotient, since all these cases are twisting cases: for D = 6, 10, 22, HD ∼= ￿ −D,2 Q ￿ . More- over, if we consider the subset QD of A2 consisting of those principally polarized abelian surfaces [(A,L)] such that O ￿→ End(A), it is obvious that this set con- tains the image of the forgetful map. The forgetful map depends on the choice of ρ and QD can have multiple components coming from these different choices. Rotger gives a formula for the number of irreducible components of QD, which in all the cases D = 6, 10, 22 turns out to be 1, that is, the set QD is irreducible, and therefore the image of the above map is the whole quaternionic locus. Finally, for a given D, we will briefly introduce the embedding ￿D : H→ H2 compatible with the actions of Γ(D, 1) and Sp(4,Z) through an embedding O(D, 1)1 ￿→ Sp(4,Z). The procedure can be found, for example, in [Has95, BG05]. Consider the non-degenerate alternate symmetric form on HD given by T (α,β) = 1 D tr(ραβ), which takes integral values on O(D, 1). Now, let us consider a symplectic basis of O(D, 1) with respect to T and denote it by {η1, . . . , η4}. Given an element γ ∈ O(D, 1)1, note that {η1γ, η2γ, η3γ, η4γ} is also a symplectic basis of O(D, 1) with respect to T . Therefore, there exists a matrix Mγ ∈ Sp(4,Z) such that {η1, η2, η3, η4}γ = {η1, η2, η3, η4}tMγ . Given τ ∈ H, consider the matrices Ω1(τ) = ￿ Φ(η1) ￿ τ 1 ￿ ,Φ(η2) ￿ τ 1 ￿￿ , Ω2(τ) = ￿ Φ(η3) ￿ τ 1 ￿ ,Φ(η4) ￿ τ 1 ￿￿ and Ω(τ) = Ω2(τ)−1 · Ω1(τ). Proposition 3.11. The map ￿D : H → H2, τ ￿→ Ω(τ), is an holomorphic 96 The curves X(D, 1) of genus 0 embedding compatible with the actions of Γ(D, 1) and Sp(4,Z) through the em- bedding O(D, 1)1 → Sp(4,Z), γ ￿→Mγ . Since this embedding does depend on several choices of elements of O(D, 1), let us fix all these necessary elements and the resulting embeddings in the cases D = 6, 10, 22 for the orders which in these cases we have already fixed in Sec- tion 3.1. We choose the values of ρ and a corresponding symplectic basis as follows: D ρ η1 η2 η3 η4 6 (0, 1, 3, 0)6 (1, 0, 0, 0)6 (−1, 0,−1, 1)6 (0, 0,−1, 0)6 (−1, 1, 0, 0)6 10 (0, 1, 0,−2)10 (1, 0, 0, 0)10 (0, 1, 0, 0)10 (0, 0, 0,−1)10 (0, 0,−1, 0)10 22 (0, 3, 11, 0)22 (1, 0, 0, 0)22 (0, 1, 3, 0)22 (0, 0, 1, 0)22 (−2, 0, 1, 1)22 With these choices, the explicit formulas for the embeddings ￿D : H → H2, where ￿D(τ) = ￿ A(τ) B(τ) B(τ) C(τ) ￿ , are the following: D A(τ) B(τ) 6 1+ √ 3−2√3τ+(1−√3)τ2 1−√3−2√3τ+(1+√3)τ2 2(1+τ2) 1−√3−2√3τ+(1+√3)τ2 10 − (1−5τ2) √ 2 2(1+6τ+5τ2) 1+2τ+5τ2 1+6τ+5τ2 22 − 3+ √ 11+2 √ 11τ−(−3+ √ 11)τ2 −3−√11+2√11τ+(−3+ √ 11)τ2 2(1+τ2) −3−√11+2√11τ+(−3+ √ 11)τ2 D C(τ) 6 −4 √ 3τ 1−√3−2√3τ+(1+√3)τ2 10 − √ 2(1−5τ2) 1+6τ+5τ2 22 2 √ 11(−1+τ2) −3−√11+2√11τ+(−3+ √ 11)τ2 Finally, we will define a set of coordinates in Sp(4,Z)\H2 and some addi- tional functions we will need when dealing with the case D = 22. 3.5 Expression of t+D in terms of absolute Igusa invariants 97 Definition. For mt = ￿ m￿ m￿￿ ￿ ∈ {0, 1}4, we define the function ϑm : H2 → C such that for every τ ∈ H2, ϑm(τ) = ￿ n∈Z2 eπi((n+m ￿/2)t·τ ·(n+m￿/2)+m￿￿·nt) which is classically called the Thetanullwerte (or thetaconstant) of characteristic m. The characteristic is said to be even (resp. odd) ifm￿t ·m￿￿ is even (resp. odd). We introduce as well the Thetanullwerte transformation formula, which we will use next to write a set of generators of the ring of Siegel modular forms of genus 2. Theorem 3.2 (Thetanullwerte transformation formula). Consider τ ∈ H2, M = ￿ A B C D ￿ ∈ Sp(4,Z) and m = ￿ m￿ m￿￿ ￿t . Then, ϑMm(Mτ) = ν(M)e 2πiφm(M)|Cτ +D|1/2ϑm(τ), where • (Mm)t = ￿ D −C −B A ￿ mt + ￿ (CDt)∆ (ABt)∆ ￿ , • ν(M)4 = eπi tr(BCt), • φm(M) = − 18 (m￿tBtDm￿ + m￿￿tAtCm￿￿ − 2m￿tBtCm￿￿ − 2(ABt)∆(Dm￿ − Cm￿￿)), • (·)∆ denotes the diagonal written as a column vector. Theorem 3.3 ([Igu62, Igu67]). The graded ring of Siegel modular forms of even weight and genus 2 is C[ψ4,ψ6,χ10,χ12], where 22ψ4 = ￿ m even ϑ8m, 22ψ6 = ￿ M∈Γ2/Γ2(2) M(ϑm1ϑm2ϑm3) 4, 98 The curves X(D, 1) of genus 0 −214χ10 = ￿ m even ϑ2m, 2173χ12 = 2 −153−411−1(23ψ26 − 2211ψ34 + 32 ￿ m even ϑ24m ). The whole graded ring of Siegel modular forms is obtained adding a form of weight 35, which can be written as well in terms of Thetanullwerte. In this way, we obtain the (Baily-Borel-)Satake compactification of the mod- uli space of principally polarized abelian surfaces, ProjC[ψ4,ψ6,χ10,χ12] ￿ PC(2, 3, 5, 6), and the moduli space of genus 2 curves can be identified with the subspace ProjC[ψ4,ψ6,χ10,χ12]￿{χ10 = 0}. Moreover, similarly to what happens in the case of elliptic curves and classical modular forms, the values of the above Siegel modular forms can be related to values of projective invariants of sextic forms. Given a sextic form u(X) = u0X6+u1X5+ · · ·+u6, denote by ξ1, . . . , ξ6 his roots and write (ij) = ξi − ξj . Then we define the Igusa’s projective invariants A,B,C,D through their irrational expressions as: A(u) = u20 ￿ fifteen (12)2(34)2(56)2, B(u) = u40 ￿ ten (12)2(23)2(31)2(45)2(56)2(64)2, C(u) = u60 ￿ sixty (12)2(23)2(31)2(45)2(56)2(64)2(14)2(25)2(36)2, D(u) = u100 ￿ j 1. 115 116 Uniformization of some Shimura curves with full Atkin-Lehner quotient of genus 0 4.1 All the curves X(D,N) with an Atkin-Lehner quotient of genus 0 The method we used to uniformize the Shimura curves in the previous chapter was not really dependent on the genus of the curvesX(D, 1) but on the existence of a genus 0 quotient. As we have noted previously, the list of Shimura curves X(D,N) of genus 0 only consists of those three curves we have already studied, but the list of Shimura curves having an Atkin-Lehner quotient of genus 0 is much longer. In order to compute a list of such curves, it is worth recalling that, since we know the genus of X(D,N), Riemann-Hurwitz formula allows us to compute the genus of the full Atkin-Lehner group once we know the set of ramification points of the morphism X(D,N) → X(D,N)+ and the corre- sponding ramification indices. However, a point P ∈ X(D,N) is a ramification point if and only if it is fixed by ω ∈ WDN ￿ {id} and the ramification index corresponding to this point is exactly the number of transformations in WDN fixing it. Therefore, we obtain all this information from the following result by Ogg (cf. [Ogg83]). Theorem 4.1. The fixed points by the Atkin-Lehner involution of level m￿DN acting on X(D,N) are CM points by Z[i],Z[ √−2], if m = 2, Z[ 1+ √−m 2 ],Z[ √−m], if m ≡ 3 (mod 4), Z[ √−m], otherwise. The number of fixed points corresponding to any of the previous orders, R, is given by h(R) ￿ p|DNm νp(O(D,N), R;O(D,N)∗), where the values of νp(O(D,N), R;O(D,N)∗) can be found in Theorem 1.7. Note that the sets of fixed points by different non-trivial Atkin-Lehner in- volutions are disjoint. Thus, the ramification index of any ramified point is 2. We list on the following table the Shimura curves X(D,N) having its full Atkin-Lehner quotient of genus 0. The values of D,N for squarefree N , which amount to 260 out of the 271 total cases were computed in [LMR06]. The rest, to our knowledge, had not been computed before. Let us briefly explain the contents of the table: 4.1 All the curves X(D,N) with an Atkin-Lehner quotient of genus 0 117 • 1st column: value of D. • 2nd column: value of N . • 3rd column: genus of X(D,N). • 4th column: number of elliptic cycles on the quotient ￿Γ(D,N),WDN ￿\H, nW . As we have seen, this determines the complexity of the procedure we have carried out in the previous chapter. • 5th and 6th columns: number of elliptic cycles in Γ(D,N)\H of order 2 and 3, e2, e3 respectively. • 7th column: This column contains a list of expressions of the form {m,Lm} where m is a divisor of DN such that gcd(m,DN/m) = 1 and Lm is a list containing the number of fixed points of the Atkin-Lehner involution. If ωm has no fixed points, we omit the term altogether. As we have seen before, if m > 2 the fixed points by ωm are CM points by certain orders inside Q( √−m). We list the number of fixed points corresponding to each order separately and ordered in function of the absolute value of the corresponding discriminants. In the case m = 2 we list first the number of fixed points corresponding to CM(Z[i]) and then those corresponding to CM(Z[ √−2]). Proposition 4.1. The following table contains all the pairs of integers D > 1, N ≥ 1 with (D,N) = 1 giving rise to Shimura curves X(D,N)+ of genus 0: D N g nW e2 e3 {m, fixed points by ωm} 6 1 0 3 2 2 {2, {2, 0}}, {3, {2, 0}}, {6, {2}} 5 1 4 4 0 {2, {4, 0}}, {6, {4}}, {10, {4}}, {30, {4}} 7 1 4 0 4 {3, {4, 0}}, {6, {4}}, {21, {4}}, {42, {4}} 11 3 5 0 0 {6, {4}}, {22, {4}}, {33, {4}}, {66, {8}} 13 1 4 4 4 {2, {4, 0}}, {3, {4, 0}}, {13, {4}}, {78, {4}} 17 3 5 4 0 {2, {4, 0}}, {34, {8}}, {51, {4, 0}}, {102, {4}} 19 3 5 0 4 {3, {4, 0}}, {19, {4, 0}}, {57, {4}}, {114, {8}} 23 5 6 0 0 {46, {8}}, {69, {8}}, {138, {8}} 25 5 6 4 0 {2, {4, 0}}, {6, {4}}, {25, {4}}, {75, {4, 0}}, {150, {8}} 118 Uniformization of some Shimura curves with full Atkin-Lehner quotient of genus 0 D N g nW e2 e3 {m, fixed points by ωm} 6 29 5 6 4 0 {2, {4, 0}}, {6, {4}}, {58, {4}}, {174, {12}} 31 5 6 0 4 {3, {4, 0}}, {6, {4}}, {93, {4}}, {186, {12}} 35 9 6 0 0 {6, {8}}, {10, {8}}, {21, {8}}, {70, {8}}, {105, {8}}, {210, {8}} 37 5 6 4 4 {2, {4, 0}}, {3, {4, 0}}, {37, {4}}, {222, {12}} 41 7 7 4 0 {2, {4, 0}}, {82, {8}}, {123, {4, 0}}, {246, {12}} 43 7 7 0 4 {3, {4, 0}}, {43, {4, 0}}, {129, {12}}, {258, {8}} 47 9 8 0 0 {94, {16}}, {141, {8}}, {282, {8}} 49 9 8 0 4 {3, {4, 0}}, {6, {4}}, {49, {8}}, {147, {4, 0}}, {294, {12}} 53 9 8 4 0 {2, {4, 0}}, {6, {4}}, {106, {12}}, {318, {12}} 55 13 7 0 0 {6, {8}}, {10, {8}}, {30, {8}}, {66, {16}}, {165, {8}}, {330, {8}} 59 11 9 0 0 {6, {4}}, {118, {12}}, {177, {4}}, {354, {16}} 61 9 8 4 4 {2, {4, 0}}, {3, {4, 0}}, {61, {12}}, {366, {12}} 65 13 7 8 0 {2, {8, 0}}, {10, {8}}, {30, {8}}, {130, {8}}, {195, {8, 0}}, {390, {16}} 67 11 9 0 4 {3, {4, 0}}, {67, {4, 0}}, {201, {12}}, {402, {16}} 71 13 10 0 0 {142, {8}}, {213, {8}}, {426, {24}} 73 11 9 4 4 {2, {4, 0}}, {3, {4, 0}}, {6, {4}}, {73, {8}}, {219, {8, 0}}, {438, {8}} 77 17 8 0 0 {6, {8}}, {21, {8}}, {33, {8}}, {66, {16}}, {154, {16}}, {462, {8}} 79 13 10 0 4 {3, {4, 0}}, {6, {4}}, {237, {12}}, {474, {20}} 83 15 11 0 0 {6, {4}}, {166, {20}}, {249, {12}}, {498, {8}} 85 17 8 8 0 {2, {8, 0}}, {30, {8}}, {34, {16}}, {51, {8, 0}}, {85, {8}}, {510, {16}} 89 15 11 4 0 {2, {4, 0}}, {178, {16}}, {267, {4, 0}}, {534, {20}} 4.1 All the curves X(D,N) with an Atkin-Lehner quotient of genus 0 119 D N g nW e2 e3 {m, fixed points by ωm} 6 91 17 8 0 8 {3, {8, 0}}, {13, {8}}, {42, {8}}, {91, {8, 0}}, {273, {8}}, {546, {24}} 95 21 9 0 0 {10, {8}}, {19, {8, 0}}, {114, {16}}, {190, {8}}, {285, {16}}, {570, {16}} 97 15 11 4 4 {2, {4, 0}}, {3, {4, 0}}, {6, {4}}, {97, {8}}, {291, {8, 0}}, {582, {16}} 107 19 13 0 0 {6, {4}}, {214, {12}}, {321, {20}}, {642, {16}} 109 17 12 4 4 {2, {4, 0}}, {3, {4, 0}}, {109, {12}}, {654, {28}} 113 19 13 4 0 {2, {4, 0}}, {226, {16}}, {339, {12, 0}}, {678, {20}} 115 25 10 0 0 {10, {8}}, {30, {8}}, {46, {16}}, {69, {16}}, {115, {8, 0}}, {345, {8}}, {690, {16}} 119 25 10 0 0 {21, {8}}, {34, {16}}, {42, {8}}, {238, {16}}, {357, {8}}, {714, {24}} 137 23 15 4 0 {2, {4, 0}}, {274, {24}}, {411, {12, 0}}, {822, {20}} 143 29 11 0 0 {13, {8}}, {22, {8}}, {66, {16}}, {286, {24}}, {429, {16}}, {858, {16}} 145 29 11 8 0 {2, {8, 0}}, {6, {8}}, {30, {8}}, {145, {16}}, {174, {24}}, {435, {8, 0}}, {870, {16}} 155 33 12 0 0 {6, {8}}, {30, {8}}, {186, {24}}, {310, {16}}, {465, {16}}, {930, {24}} 161 33 12 0 0 {21, {8}}, {42, {8}}, {69, {16}}, {138, {16}}, {322, {16}}, {483, {8, 0}}, {966, {24}} 203 41 14 0 0 {6, {8}}, {42, {8}}, {174, {24}}, {406, {32}}, {609, {16}}, {1218, {24}} 235 49 16 0 0 {10, {8}}, {30, {8}}, {94, {32}}, {141, {16}}, {235, {8, 0}}, {705, {24}}, {1410, {32}} 10 1 0 4 0 4 {2, {0, 2}}, {5, {2}}, {10, {2}} 3 1 4 0 4 {2, {0, 4}}, {3, {4, 0}}, {5, {4}}, {30, {4}} 7 1 5 0 8 {5, {4}}, {10, {4}}, {35, {4, 0}}, {70, {4}} 120 Uniformization of some Shimura curves with full Atkin-Lehner quotient of genus 0 D N g nW e2 e3 {m, fixed points by ωm} 10 9 5 6 0 0 {2, {0, 4}}, {5, {4}}, {18, {4}}, {45, {4}}, {90, {8}} 11 5 6 0 0 {2, {0, 4}}, {10, {4}}, {22, {4}}, {110, {12}} 13 3 6 0 8 {10, {4}}, {13, {4}}, {65, {8}}, {130, {4}} 17 7 7 0 0 {2, {0, 4}}, {17, {8}}, {85, {4}}, {170, {12}} 19 5 7 0 8 {2, {0, 4}}, {10, {4}}, {38, {12}}, {190, {4}} 21 9 6 0 8 {3, {8, 0}}, {5, {8}}, {35, {8, 0}}, {42, {8}}, {105, {8}}, {210, {8}} 23 9 8 0 0 {5, {4}}, {10, {4}}, {115, {4, 0}}, {230, {20}} 29 11 9 0 0 {5, {4}}, {58, {4}}, {145, {8}}, {290, {20}} 31 9 9 0 8 {62, {16}}, {155, {8, 0}}, {310, {8}} 33 17 8 0 0 {2, {0, 8}}, {30, {8}}, {33, {8}}, {110, {24}}, {165, {8}}, {330, {8}} 37 11 10 0 8 {10, {4}}, {37, {4}}, {185, {16}}, {370, {12}} 39 17 8 0 8 {3, {8, 0}}, {30, {8}}, {65, {16}}, {78, {8}}, {195, {8, 0}}, {390, {16}} 41 15 11 0 0 {2, {0, 4}}, {5, {4}}, {10, {4}}, {82, {8}}, {205, {8}}, {410, {16}} 49 17 13 0 8 {5, {4}}, {10, {4}}, {98, {16}}, {245, {12}}, {490, {12}} 51 25 10 0 0 {2, {0, 8}}, {17, {16}}, {30, {8}}, {102, {8}}, {170, {24}}, {510, {16}} 53 19 13 0 0 {10, {4}}, {53, {12}}, {265, {8}}, {530, {28}} 57 25 10 0 8 {2, {0, 8}}, {3, {8, 0}}, {38, {24}}, {57, {8}}, {285, {16}}, {570, {16}} 61 19 14 0 8 {5, {4}}, {122, {20}}, {305, {16}}, {610, {12}} 69 33 12 0 0 {5, {8}}, {30, {8}}, {138, {16}}, {230, {40}}, {345, {8}}, {690, {16}} 77 33 12 0 0 {10, {8}}, {35, {8, 0}}, {77, {16}}, {110, {24}}, {385, {8}}, {770, {32}} 93 41 14 0 8 {3, {8, 0}}, {30, {8}}, {62, {32}}, {93, {8}}, {155, {16, 0}}, {465, {16}}, {930, {24}} 4.1 All the curves X(D,N) with an Atkin-Lehner quotient of genus 0 121 D N g nW e2 e3 {m, fixed points by ωm} 14 1 1 4 2 0 {2, {2, 2}}, {14, {4}} 3 3 5 0 0 {2, {0, 4}}, {14, {8}}, {21, {4}}, {42, {4}} 5 3 5 4 0 {2, {4, 0}}, {14, {8}}, {35, {4, 0}}, {70, {4}} 9 7 7 0 0 {2, {0, 4}}, {9, {4}}, {14, {8}}, {18, {4}}, {126, {8}} 11 7 7 0 0 {2, {0, 4}}, {11, {4, 0}}, {22, {4}}, {77, {8}}, {154, {8}} 13 7 7 4 0 {2, {4, 0}}, {14, {8}}, {91, {4, 0}}, {182, {12}} 15 13 7 0 0 {14, {16}}, {21, {8}}, {30, {8}}, {35, {8, 0}}, {105, {8}}, {210, {8}} 19 11 9 0 0 {2, {0, 4}}, {14, {8}}, {133, {4}}, {266, {20}} 23 13 10 0 0 {14, {8}}, {46, {8}}, {161, {16}}, {322, {8}} 25 15 11 4 0 {2, {4, 0}}, {14, {8}}, {25, {4}}, {50, {12}}, {350, {16}} 29 15 11 4 0 {2, {4, 0}}, {29, {12}}, {58, {4}}, {203, {8, 0}}, {406, {16}} 37 19 13 4 0 {2, {4, 0}}, {37, {4}}, {74, {20}}, {259, {8, 0}}, {518, {16}} 39 29 11 0 0 {14, {16}}, {42, {8}}, {78, {8}}, {182, {24}}, {273, {8}}, {546, {24}} 43 23 15 0 0 {2, {0, 4}}, {43, {4, 0}}, {86, {20}}, {301, {8}}, {602, {24}} 57 41 14 0 0 {2, {0, 8}}, {14, {16}}, {21, {8}}, {57, {8}}, {114, {16}}, {266, {40}}, {798, {16}} 15 1 1 4 0 2 {3, {2, 2}}, {15, {2, 2}} 2 3 5 0 0 {3, {0, 4}}, {10, {4}}, {15, {4, 4}}, {30, {4}} 4 5 6 0 0 {3, {0, 4}}, {12, {4}}, {15, {4, 8}}, {60, {4}} 7 5 6 0 4 {3, {4, 4}}, {7, {4, 4}}, {105, {8}} 8 9 8 0 0 {15, {4, 12}}, {40, {8}}, {120, {8}} 11 9 8 0 0 {33, {8}}, {55, {8, 8}}, {165, {8}} 13 9 8 0 4 {3, {4, 4}}, {13, {8}}, {195, {4, 12}} 122 Uniformization of some Shimura curves with full Atkin-Lehner quotient of genus 0 D N g nW e2 e3 {m, fixed points by ωm} 15 14 17 8 0 0 {3, {0, 8}}, {7, {8, 8}}, {10, {8}}, {42, {8}}, {70, {8}}, {105, {8}}, {210, {8}} 17 13 10 0 0 {15, {4, 4}}, {85, {8}}, {255, {12, 12}} 19 13 10 0 4 {3, {4, 4}}, {15, {4, 4}}, {57, {8}}, {285, {16}} 22 25 10 0 0 {10, {8}}, {22, {8}}, {30, {8}}, {33, {8}}, {55, {16, 16}}, {165, {8}}, {330, {8}} 26 29 11 0 0 {3, {0, 8}}, {10, {8}}, {13, {8}}, {30, {8}}, {78, {8}}, {130, {8}}, {195, {0, 24}}, {390, {16}} 29 21 14 0 0 {87, {12, 12}}, {145, {16}}, {435, {4, 12}} 34 37 13 0 0 {15, {8, 8}}, {30, {8}}, {85, {8}}, {102, {8}}, {255, {24, 24}}, {510, {16}} 41 29 18 0 0 {123, {4, 12}}, {205, {16}}, {615, {20, 20}} 21 1 1 5 4 0 {7, {2, 2}}, {21, {4}} 2 3 5 4 0 {2, {4, 0}}, {7, {4, 4}}, {21, {4}}, {42, {4}} 4 7 7 0 0 {4, {4}}, {7, {4, 8}}, {28, {4}}, {84, {8}} 5 5 7 8 0 {15, {4, 4}}, {21, {8}}, {105, {8}} 10 17 8 8 0 {2, {8, 0}}, {15, {8, 8}}, {21, {8}}, {30, {8}}, {70, {8}}, {105, {8}}, {210, {8}} 11 13 10 0 0 {7, {4, 4}}, {21, {8}}, {231, {12, 12}} 13 13 11 8 0 {39, {8, 8}}, {91, {4, 12}}, {273, {8}} 19 21 14 0 0 {21, {8}}, {57, {8}}, {133, {8}}, {399, {16, 16}} 22 37 13 0 0 {7, {8, 8}}, {21, {8}}, {22, {8}}, {154, {16}}, {231, {24, 24}}, {462, {8}} 26 41 14 8 0 {2, {8, 0}}, {39, {16, 16}}, {42, {8}}, {78, {8}}, {91, {0, 24}}, {273, {8}}, {546, {24}} 22 1 0 4 2 4 {2, {2, 0}}, {11, {2, 0}}, {22, {2}} 3 3 5 0 4 {3, {4, 0}}, {11, {4, 0}}, {33, {4}}, {66, {8}} 5 5 6 4 0 {2, {4, 0}}, {5, {4}}, {11, {4, 0}}, {110, {12}} 7 5 7 0 8 {14, {8}}, {77, {8}}, {154, {8}} 4.1 All the curves X(D,N) with an Atkin-Lehner quotient of genus 0 123 D N g nW e2 e3 {m, fixed points by ωm} 22 13 9 9 4 8 {2, {4, 0}}, {22, {4}}, {26, {12}}, {286, {12}} 15 21 9 0 0 {5, {8}}, {11, {8, 0}}, {66, {16}}, {110, {24}}, {165, {8}}, {330, {8}} 17 15 11 4 0 {2, {4, 0}}, {34, {8}}, {187, {4, 0}}, {374, {28}} 19 15 12 0 8 {22, {4}}, {38, {12}}, {209, {20}}, {418, {8}} 21 25 10 0 8 {3, {8, 0}}, {14, {16}}, {33, {8}}, {42, {8}}, {66, {16}}, {77, {16}}, {462, {8}} 35 41 14 0 0 {5, {8}}, {14, {16}}, {70, {8}}, {110, {24}}, {154, {16}}, {385, {8}}, {770, {32}} 26 1 2 5 0 0 {2, {0, 2}}, {13, {2}}, {26, {6}} 3 5 6 0 0 {2, {0, 4}}, {6, {4}}, {26, {12}}, {78, {4}} 5 7 7 0 0 {5, {4}}, {26, {12}}, {65, {8}}, {130, {4}} 7 9 8 0 0 {13, {4}}, {26, {12}}, {91, {4, 0}}, {182, {12}} 9 13 10 0 0 {2, {0, 4}}, {18, {4}}, {26, {12}}, {117, {8}}, {234, {12}} 15 25 10 0 0 {5, {8}}, {6, {8}}, {26, {24}}, {65, {16}}, {195, {8, 0}}, {390, {16}} 17 19 13 0 0 {2, {0, 4}}, {13, {4}}, {26, {12}}, {34, {8}}, {221, {16}}, {442, {8}} 21 33 12 0 0 {6, {8}}, {21, {8}}, {26, {24}}, {182, {24}}, {273, {8}}, {546, {24}} 33 1 1 5 4 2 {3, {2, 2}}, {33, {4}} 2 5 6 4 0 {2, {4, 0}}, {3, {0, 4}}, {22, {4}}, {33, {4}}, {66, {8}} 5 9 9 8 0 {15, {4, 4}}, {55, {8, 8}}, {165, {8}} 7 13 10 0 4 {3, {4, 4}}, {33, {8}}, {231, {12, 12}} 10 29 11 8 0 {2, {8, 0}}, {15, {8, 8}}, {55, {16, 16}}, {66, {16}}, {165, {8}}, {330, {8}} 14 41 14 0 0 {3, {0, 8}}, {33, {8}}, {42, {8}}, {66, {16}}, {154, {16}}, {231, {24, 24}}, {462, {8}} 34 1 1 5 0 4 {17, {4}}, {34, {4}} 124 Uniformization of some Shimura curves with full Atkin-Lehner quotient of genus 0 D N g nW e2 e3 {m, fixed points by ωm} 34 3 5 6 0 4 {3, {4, 0}}, {6, {4}}, {17, {8}}, {51, {4, 0}}, {102, {4}} 5 9 8 0 0 {5, {4}}, {10, {4}}, {34, {8}}, {85, {4}}, {170, {12}} 7 9 9 0 8 {14, {8}}, {17, {8}}, {34, {8}}, {238, {8}} 11 17 12 0 0 {11, {4, 0}}, {17, {8}}, {22, {4}}, {187, {4, 0}}, {374, {28}} 35 1 3 6 0 0 {7, {2, 2}}, {35, {2, 6}} 2 7 7 0 0 {2, {0, 4}}, {7, {4, 4}}, {35, {0, 12}}, {70, {4}} 3 9 8 0 0 {15, {4, 4}}, {35, {4, 12}}, {105, {8}} 4 13 10 0 0 {7, {4, 8}}, {28, {4}}, {35, {0, 12}}, {140, {12}} 6 25 10 0 0 {2, {0, 8}}, {15, {8, 8}}, {30, {8}}, {35, {0, 24}}, {42, {8}}, {105, {8}}, {210, {8}} 13 29 18 0 0 {35, {4, 12}}, {65, {16}}, {455, {20, 20}} 38 1 2 5 2 0 {2, {2, 0}}, {19, {2, 0}}, {38, {6}} 3 7 7 0 0 {6, {4}}, {38, {12}}, {57, {4}}, {114, {8}} 7 13 10 0 0 {19, {4, 0}}, {38, {12}}, {133, {4}}, {266, {20}} 13 21 14 4 0 {2, {4, 0}}, {26, {12}}, {38, {12}}, {494, {28}} 39 1 3 6 0 0 {13, {4}}, {39, {4, 4}} 2 7 7 0 0 {6, {4}}, {13, {4}}, {39, {8, 8}}, {78, {4}} 4 13 10 0 0 {39, {8, 16}}, {52, {8}}, {156, {8}} 5 13 10 0 0 {15, {4, 4}}, {39, {8, 8}}, {195, {4, 12}} 7 17 12 0 0 {7, {4, 4}}, {13, {8}}, {21, {8}}, {91, {4, 12}}, {273, {8}} 10 37 13 0 0 {6, {8}}, {15, {8, 8}}, {39, {16, 16}}, {130, {8}}, {195, {0, 24}}, {390, {16}} 46 1 1 5 2 4 {2, {2, 2}}, {46, {4}} 3 7 7 0 4 {2, {0, 4}}, {3, {4, 0}}, {6, {4}}, {69, {8}}, {138, {8}} 4.1 All the curves X(D,N) with an Atkin-Lehner quotient of genus 0 125 D N g nW e2 e3 {m, fixed points by ωm} 46 5 11 9 4 0 {2, {4, 0}}, {46, {8}}, {115, {4, 0}}, {230, {20}} 51 1 3 6 0 2 {3, {2, 2}}, {51, {2, 6}} 2 9 8 0 0 {3, {0, 4}}, {6, {4}}, {34, {8}}, {51, {0, 12}}, {102, {4}} 5 17 12 0 0 {51, {4, 12}}, {85, {8}}, {255, {12, 12}} 10 49 16 0 0 {6, {8}}, {10, {8}}, {34, {16}}, {51, {0, 24}}, {85, {8}}, {255, {24, 24}}, {510, {16}} 55 1 3 7 0 4 {5, {4}}, {55, {4, 4}} 2 11 9 0 0 {5, {4}}, {22, {4}}, {55, {8, 8}}, {110, {12}} 3 13 10 0 4 {3, {4, 4}}, {5, {8}}, {15, {4, 4}}, {33, {8}}, {165, {8}} 57 1 3 7 4 0 {19, {2, 6}}, {57, {4}} 2 9 8 4 0 {2, {4, 0}}, {6, {4}}, {19, {0, 12}}, {57, {4}}, {114, {8}} 7 25 16 0 0 {7, {4, 4}}, {19, {4, 12}}, {133, {8}}, {399, {16, 16}} 58 1 2 6 0 4 {2, {0, 2}}, {29, {6}}, {58, {2}} 3 9 8 0 4 {2, {0, 4}}, {3, {4, 0}}, {29, {12}}, {174, {12}} 5 15 11 0 0 {10, {4}}, {29, {12}}, {145, {8}}, {290, {20}} 62 1 3 6 2 0 {2, {2, 2}}, {62, {8}} 3 11 9 0 0 {2, {0, 4}}, {62, {16}}, {93, {4}}, {186, {12}} 7 21 14 0 0 {14, {8}}, {62, {16}}, {217, {8}}, {434, {24}} 65 1 5 8 0 0 {5, {4}}, {13, {4}}, {65, {8}} 3 17 12 0 0 {5, {8}}, {15, {4, 4}}, {65, {16}}, {195, {4, 12}} 69 1 3 7 4 2 {3, {2, 2}}, {69, {8}} 2 11 9 4 0 {2, {4, 0}}, {3, {0, 4}}, {6, {4}}, {46, {8}}, {69, {8}}, {138, {8}} 74 1 4 7 0 0 {2, {0, 2}}, {37, {2}}, {74, {10}} 3 13 10 0 0 {2, {0, 4}}, {6, {4}}, {74, {20}}, {222, {12}} 126 Uniformization of some Shimura curves with full Atkin-Lehner quotient of genus 0 D N g nW e2 e3 {m, fixed points by ωm} 74 5 19 13 0 0 {5, {4}}, {74, {20}}, {185, {16}}, {370, {12}} 77 1 5 9 4 0 {11, {2, 6}}, {77, {8}} 2 15 11 4 0 {2, {4, 0}}, {11, {0, 12}}, {14, {8}}, {22, {4}}, {77, {8}}, {154, {8}} 3 21 14 0 0 {11, {4, 12}}, {77, {16}}, {231, {12, 12}} 82 1 3 7 0 4 {41, {8}}, {82, {4}} 3 13 10 0 4 {3, {4, 0}}, {6, {4}}, {41, {16}}, {123, {4, 0}}, {246, {12}} 85 1 5 9 0 4 {5, {4}}, {17, {8}}, {85, {4}} 3 21 14 0 4 {3, {4, 4}}, {5, {8}}, {17, {16}}, {255, {12, 12}} 86 1 4 7 2 0 {2, {2, 0}}, {43, {2, 0}}, {86, {10}} 3 15 11 0 0 {6, {4}}, {86, {20}}, {129, {12}}, {258, {8}} 87 1 5 8 0 2 {3, {2, 2}}, {87, {6, 6}} 2 15 11 0 0 {3, {0, 4}}, {58, {4}}, {87, {12, 12}}, {174, {12}} 93 1 5 9 4 0 {31, {6, 6}}, {93, {4}} 2 15 11 4 0 {2, {4, 0}}, {31, {12, 12}}, {93, {4}}, {186, {12}} 94 1 3 7 2 4 {2, {2, 2}}, {94, {8}} 95 1 7 10 0 0 {5, {4}}, {95, {8, 8}} 2 19 13 0 0 {5, {4}}, {38, {12}}, {95, {16, 16}}, {190, {4}} 3 25 16 0 0 {5, {8}}, {57, {8}}, {95, {16, 16}}, {285, {16}} 106 1 4 8 0 4 {2, {0, 2}}, {53, {6}}, {106, {6}} 111 1 7 10 0 0 {37, {4}}, {111, {8, 8}} 2 19 13 0 0 {6, {4}}, {37, {4}}, {111, {16, 16}}, {222, {12}} 115 1 7 11 0 4 {23, {6, 6}}, {115, {2, 6}} 2 23 15 0 0 {2, {0, 4}}, {23, {12, 12}}, {115, {0, 12}}, {230, {20}} 118 1 4 8 2 4 {2, {2, 0}}, {59, {6, 0}}, {118, {6}} 4.1 All the curves X(D,N) with an Atkin-Lehner quotient of genus 0 127 D N g nW e2 e3 {m, fixed points by ωm} 119 1 9 12 0 0 {7, {2, 2}}, {119, {10, 10}} 2 25 16 0 0 {7, {4, 4}}, {14, {8}}, {119, {20, 20}}, {238, {8}} 122 1 6 9 0 0 {2, {0, 2}}, {61, {6}}, {122, {10}} 129 1 7 11 4 0 {43, {2, 6}}, {129, {12}} 134 1 6 9 2 0 {2, {2, 0}}, {67, {2, 0}}, {134, {14}} 3 23 15 0 0 {6, {4}}, {134, {28}}, {201, {12}}, {402, {16}} 143 1 11 14 0 0 {11, {2, 6}}, {143, {10, 10}} 2 31 19 0 0 {11, {0, 12}}, {26, {12}}, {143, {20, 20}}, {286, {12}} 146 1 7 10 0 0 {73, {4}}, {146, {16}} 159 1 9 12 0 2 {3, {2, 2}}, {159, {10, 10}} 2 27 17 0 0 {3, {0, 4}}, {106, {12}}, {159, {20, 20}}, {318, {12}} 161 1 11 15 4 0 {23, {6, 6}}, {161, {16}} 166 1 6 10 2 4 {2, {2, 0}}, {83, {6, 0}}, {166, {10}} 178 1 7 11 0 4 {89, {12}}, {178, {8}} 183 1 11 14 0 0 {61, {12}}, {183, {8, 8}} 194 1 9 12 0 0 {97, {4}}, {194, {20}} 202 1 8 12 0 4 {2, {0, 2}}, {101, {14}}, {202, {6}} 206 1 9 12 2 0 {2, {2, 2}}, {206, {20}} 210 1 5 5 0 0 {30, {8}}, {42, {8}}, {70, {8}}, {105, {8}}, {210, {8}} 11 49 10 0 0 {22, {16}}, {30, {16}}, {105, {16}}, {165, {16}}, {330, {16}}, {385, {16}}, {462, {16}}, {1155, {16, 0}}, {2310, {32}} 13 57 11 0 0 {30, {16}}, {42, {16}}, {78, {16}}, {105, {16}}, {130, {16}}, {273, {16}}, {910, {32}}, {1365, {16}}, {2730, {32}} 17 73 13 0 0 {30, {16}}, {42, {16}}, {70, {16}}, {85, {16}}, {102, {16}}, {238, {32}}, {357, {16}}, {595, {16, 0}}, {1785, {32}}, {3570, {32}} 128 Uniformization of some Shimura curves with full Atkin-Lehner quotient of genus 0 D N g nW e2 e3 {m, fixed points by ωm} 210 19 81 14 0 0 {57, {16}}, {70, {16}}, {105, {16}}, {133, {16}}, {190, {16}}, {798, {32}}, {1330, {48}}, {1995, {16, 0}}, {3990, {48}} 215 1 15 18 0 0 {43, {2, 6}}, {215, {14, 14}} 237 1 13 17 4 0 {79, {10, 10}}, {237, {12}} 314 1 14 17 0 0 {2, {0, 2}}, {157, {6}}, {314, {26}} 330 1 5 5 0 8 {3, {8, 0}}, {22, {8}}, {33, {8}}, {165, {8}}, {330, {8}} 7 49 10 0 16 {3, {16, 0}}, {33, {16}}, {42, {16}}, {70, {16}}, {210, {16}}, {385, {16}}, {462, {16}}, {1155, {16, 0}}, {2310, {32}} 13 89 15 0 16 {3, {16, 0}}, {22, {16}}, {78, {16}}, {130, {16}}, {165, {16}}, {390, {32}}, {715, {16, 0}}, {858, {32}}, {2145, {32}}, {4290, {48}} 390 1 9 6 0 0 {13, {8}}, {78, {8}}, {130, {8}}, {195, {8, 0}}, {390, {16}} 7 65 12 0 0 {13, {16}}, {70, {16}}, {195, {16, 0}}, {210, {16}}, {273, {16}}, {390, {32}}, {910, {32}}, {1365, {16}}, {2730, {32}} 462 1 9 6 8 0 {2, {8, 0}}, {22, {8}}, {42, {8}}, {154, {16}}, {462, {8}} 5 57 11 16 0 {2, {16, 0}}, {70, {16}}, {154, {32}}, {165, {16}}, {210, {16}}, {330, {16}}, {385, {16}}, {1155, {16, 0}}, {2310, {32}} 510 1 9 6 0 8 {3, {8, 0}}, {10, {8}}, {85, {8}}, {102, {8}}, {510, {16}} 7 81 14 0 16 {3, {16, 0}}, {10, {16}}, {105, {16}}, {210, {16}}, {238, {32}}, {357, {16}}, {510, {32}}, {595, {16, 0}}, {1785, {32}}, {3570, {32}} 546 1 13 7 0 0 {21, {8}}, {78, {8}}, {91, {8, 0}}, {273, {8}}, {546, {24}} 4.1 All the curves X(D,N) with an Atkin-Lehner quotient of genus 0 129 D N g nW e2 e3 {m, fixed points by ωm} 546 5 73 13 0 0 {21, {16}}, {70, {16}}, {91, {16, 0}}, {130, {16}}, {210, {16}}, {546, {48}}, {910, {32}}, {1365, {16}}, {2730, {32}} 570 1 13 7 0 0 {30, {8}}, {57, {8}}, {190, {8}}, {285, {16}}, {570, {16}} 7 97 16 0 0 {42, {16}}, {133, {16}}, {210, {16}}, {285, {32}}, {570, {32}}, {798, {32}}, {1330, {48}}, {1995, {16, 0}}, {3990, {48}} 690 1 13 7 0 8 {3, {8, 0}}, {115, {8, 0}}, {138, {16}}, {345, {8}}, {690, {16}} 714 1 17 8 0 0 {51, {8, 0}}, {102, {8}}, {238, {16}}, {357, {8}}, {714, {24}} 770 1 21 9 0 0 {22, {8}}, {70, {8}}, {77, {16}}, {385, {8}}, {770, {32}} 3 81 14 0 0 {42, {16}}, {77, {32}}, {165, {16}}, {210, {16}}, {330, {16}}, {462, {16}}, {770, {64}}, {1155, {16, 0}}, {2310, {32}} 798 1 17 8 8 0 {2, {8, 0}}, {42, {8}}, {57, {8}}, {114, {16}}, {133, {8}}, {798, {16}} 858 1 21 9 0 0 {33, {8}}, {78, {8}}, {286, {24}}, {429, {16}}, {858, {16}} 870 1 17 8 0 8 {3, {8, 0}}, {10, {8}}, {58, {8}}, {145, {16}}, {435, {8, 0}}, {870, {16}} 910 1 25 10 0 0 {2, {0, 8}}, {65, {16}}, {70, {8}}, {130, {8}}, {182, {24}}, {910, {16}} 930 1 21 9 0 0 {10, {8}}, {93, {8}}, {310, {16}}, {465, {16}}, {930, {24}} 966 1 21 9 8 0 {2, {8, 0}}, {46, {16}}, {322, {16}}, {483, {8, 0}}, {966, {24}} 1110 1 25 10 0 0 {37, {8}}, {222, {24}}, {370, {24}}, {555, {8, 0}}, {1110, {16}} 1122 1 25 10 0 8 {3, {8, 0}}, {22, {8}}, {34, {16}}, {102, {8}}, {187, {8, 0}}, {561, {16}}, {1122, {16}} 130 Uniformization of some Shimura curves with full Atkin-Lehner quotient of genus 0 D N g nW e2 e3 {m, fixed points by ωm} 1190 1 33 12 0 0 {85, {8}}, {170, {24}}, {238, {16}}, {595, {8, 0}}, {1190, {40}} 1218 1 29 11 0 0 {21, {8}}, {58, {8}}, {406, {32}}, {609, {16}}, {1218, {24}} 1230 1 25 10 0 8 {3, {8, 0}}, {30, {8}}, {82, {16}}, {123, {8, 0}}, {205, {16}}, {1230, {24}} 1254 1 29 11 8 0 {2, {8, 0}}, {66, {16}}, {114, {16}}, {418, {16}}, {627, {8, 0}}, {1254, {24}} 1290 1 29 11 0 0 {10, {8}}, {43, {8, 0}}, {258, {16}}, {430, {24}}, {645, {16}}, {1290, {16}} 1302 1 29 11 8 0 {2, {8, 0}}, {93, {8}}, {186, {24}}, {217, {16}}, {651, {16, 0}}, {1302, {16}} 1326 1 33 12 0 0 {6, {8}}, {34, {16}}, {78, {8}}, {102, {8}}, {442, {16}}, {1326, {40}} 1410 1 29 11 0 8 {3, {8, 0}}, {235, {8, 0}}, {282, {16}}, {705, {24}}, {1410, {32}} 1590 1 33 12 0 8 {3, {8, 0}}, {30, {8}}, {265, {16}}, {318, {24}}, {795, {8, 0}}, {1590, {32}} 1722 1 41 14 0 0 {123, {8, 0}}, {246, {24}}, {574, {32}}, {861, {24}}, {1722, {24}} 1770 1 37 13 0 8 {3, {8, 0}}, {118, {24}}, {177, {8}}, {885, {24}}, {1770, {40}} 1794 1 45 15 0 0 {6, {8}}, {13, {8}}, {46, {16}}, {78, {8}}, {138, {16}}, {598, {16}}, {897, {16}}, {1794, {32}} 1914 1 45 15 0 8 {3, {8, 0}}, {58, {8}}, {66, {16}}, {174, {24}}, {957, {16}}, {1914, {48}} 1938 1 49 16 0 0 {6, {8}}, {57, {8}}, {102, {8}}, {114, {16}}, {646, {32}}, {969, {24}}, {1938, {32}} 1974 1 45 15 8 0 {2, {8, 0}}, {21, {8}}, {42, {8}}, {141, {16}}, {282, {16}}, {658, {16}}, {987, {16, 0}}, {1974, {32}} 2010 1 45 15 0 0 {10, {8}}, {67, {8, 0}}, {402, {32}}, {670, {24}}, {1005, {16}}, {2010, {32}} 4.1 All the curves X(D,N) with an Atkin-Lehner quotient of genus 0 131 D N g nW e2 e3 {m, fixed points by ωm} 2130 1 45 15 0 8 {3, {8, 0}}, {10, {8}}, {30, {8}}, {142, {16}}, {213, {16}}, {355, {16, 0}}, {1065, {16}}, {2130, {32}} In order to minimize the required computations to prove that the previous table contains all the curves with maximal Atkin-Lehner quotient of genus 0, we will reduce to the squarefree level case found in [LMR06] thanks to the following lemma. Lemma 4.1. Consider the Shimura curve X(D,N)+ and write N = N ￿l2r such that l2 ￿ N ￿ for a certain prime l and an integer r > 0. If g(X(D,N)+) = 0, then g(X(D,N ￿)+) = 0. Proof. The Euler characteristic of X(D,N ￿)+ can be computed by the gen- eral formula which relates the genus, volume and order of the elliptic cycles (cf. Theorem 1.10), χ(X(D,N ￿)+) = −V (D,N ￿)+ + 1 2 (d￿2 + e ￿ 2 + 3 2 e￿4) + 2 3 e￿3 + 5 6 e￿6, where V (D,N ￿)+ is the normalized volume of the quotient Γ(D,N ￿)+\H and e￿i denotes the number of elliptic cycles of order i coming from elliptic cycles in X(D,N ￿) and d￿2 the number of elliptic cycles of order 2 which come from ordinary cycles in X(D,N ￿). Note that in such a quotient the only possible orders for elliptic points are 2, 3, 4, 6. Moreover, using the formulas for the number of elliptic cycles of orders 2 and 3 in the curveX(D,N) and the formulas for the number of fixed points by the Atkin-Lehner involutions, we obtain that e￿6 = ￿ 1, if there exist elliptic cycles of order 3 in X(D,N ￿) and 3￿DN ￿, 0, otherwise, e￿4 = ￿ 1, if there exist elliptic cycles of order 2 in X(D,N ￿) and 2￿DN ￿, 0, otherwise, e￿3 = ￿ 1, if there exist elliptic cycles of order 3 in X(D,N ￿) and 3 ￿ DN ￿, 0, otherwise, e￿2 = ￿ 1, if there exist elliptic cycles of order 2 in X(D,N ￿) and 2 ￿ DN ￿, 0, otherwise, 132 Uniformization of some Shimura curves with full Atkin-Lehner quotient of genus 0 and d￿2 = 2 −k+1 ￿ m￿DN ￿ m ￿=1 ￿ R h(R) ￿ p|DN ￿ m νp(R,O(D,N ￿)), where k denotes the number of different primes dividing DN ￿ and for each m the sum runs over the corresponding orders R described in Theorem 4.1, except for Z[i] if m = 2 and Z[ 1+ √−3 2 ] if m = 3. Observe, in particular, that e￿2 + e￿4 ≤ 1 and e￿3 + e￿6 ≤ 1 and therefore, g(X(D,N)+) = 0 if and only if 2V (D,N)+ < d￿2 + e￿2 + 3 2e ￿ 4 or equivalently 2V (D,N)+ < d￿2. We will separate the proof in two cases depending on the parity of νlN . In first place we will deal with the case of even valuation. This means that N = N ￿l2r and l ￿ N ￿. Then, χ(X(D,N)+) = −V (D,N ￿)+ψ(l2r) 2 + 1 2 (d2 + e2 + 3 2 e4) + 2 3 e3 + 5 6 e6, where ei, d2 are defined as above and can be computed by the above formulas replacing N ￿ by N . In particular ei ≤ e￿i for i = 2, 3, 4, 6 and d2 = 2 −k ￿ m|DN ￿ m>1 ￿ R h(R) ￿ p|DN￿l2rm νp(R,O(D,N)) + 2−k ￿ l2r|m|DN ￿l2r ￿ R h(R) ￿ p|DN￿l2rm νp(R,O(D,N)) ≤d￿2 + 1 2 ψ(lr)(d￿2 + 1 2 (e￿2 + e ￿ 4)) ≤d￿2 + 1 2 ψ(lr)(d￿2 + e ￿ 2 + 3 2 e￿4). Thus, if we assume that g(X(D,N)+) = 0, we obtain that 0 < −V (D,N ￿)+ψ(l 2r) 2 + 1 2 (d2 + e2 + 3 2 e4) which replacing d2, e2 and e4 by the upper bounds we have just computed, becomes 2V (D,N ￿)+ ψ(l2r) 2 + ψ(lr) < d￿2 + e ￿ 2 + 3 2 e￿4. Now observing that ψ(l 2r) 2+ψ(lr) = lrψ(lr) 2+ψ(lr) ≥ l rψ(lr) 2ψ(lr) ≥ 1, we obtain that 2V (D,N ￿)+ < d￿2 + e ￿ 2 + 3 2 e￿4, 4.1 All the curves X(D,N) with an Atkin-Lehner quotient of genus 0 133 and therefore the result follows. On the other hand, if N has odd l-adic valuation, we have N = N ￿l2r with l￿N ￿. The discussion of this case follows the same lines above, but in this case the number of distinct primes dividing both DN and DN ￿ is the same. Thus, χ(X(D,N)+) = −V (D,N ￿)+ψ(l2r) 2 + 1 2 (d2 + e2) + 2 3 e3 + 3 4 e4 + 5 6 e6, and ei ≤ e￿i for i = 2, 3, 4, 6. Similarly, d2 = 2 −k+1 ￿ m|DN m>1 (m,l)=1 ￿ R h(R) ￿ p|DN￿l2rm νp(R,O(D,N)) + 2−k+1 ￿ l2r+1|m|DN ￿l2r ￿ R h(R) ￿ p|DN￿l2rm νp(R,O(D,N)) = 2−k+1 ￿ m|DN ￿ m>1 (m,l)=1 ￿ R h(R) ￿ p|DN￿m νp(R,O(D,N ￿)) + 2−k+1ψ(lr) ￿ l|m|DN ￿ ￿ R h(R) ￿ p|DN￿m νp(R,O(D,N ￿)) ≤ 1 2 ψ(lr)d￿2. Therefore, g(X(D,N)+) = 0 if and only if V (D,N ￿)+ψ(l2r) 2 < 1 2d2 ≤ 14ψ(lr)d￿2, which immediately yields that 2V (D,N)+ < d￿2 as desired. Proof of the Proposition 4.1. The values of D,N with squarefree N are given in [LMR06]. Therefore, we only need to find the lacking non-squarefree N and to do so we can follow two different approaches, the first one would be to follow the same method used in the aforementioned article, namely that the normalized volume of such a quotient of the upper half-plane must be smaller than 64/3, which provides a fairly short list from which we can se- lect the genus 0 ones computing it in every case. However, the previous lemma allows us to restrict to pairs of the form D,N ￿(N ￿￿)2 with squarefree N ￿, and D,N ￿, now with N ￿ squarefree in the list. Next we are going to use the bound on the volume to reduce the possible values of N ￿￿ to a short list. Let us fix a couple D,N ￿ with N ￿ squarefree such that X(D,N ￿)+ has genus zero. 134 Uniformization of some Shimura curves with full Atkin-Lehner quotient of genus 0 We want to compute the integers N ￿￿ such that the curve X(D,N)+ with N = N ￿(N ￿￿)2 has genus 0 too. If we denote by V (D,N ￿)+ the normalized volume of the corresponding quotient of the upper half-plane and k the num- ber of different primes dividing N ￿￿, the volume of the quotient attached to X(D,N)+ is V (D,N)+ = ψ(N)2kψ(N ￿)V (D,N ￿)+ ≥ N ￿￿22k V (D,N ￿)+. The Selberg- Zograf bound guarantees that 64/3 ≥ V (D,N)+ ≥ N ￿￿22k V (D,N ￿)+, which gives N ￿￿ 2k/2 ≤ ￿ 64 3V (D,N ￿) , and therefore we obtain a small number of cases to check for each of the curves in [LMR06]. After carrying out these computations, we find the 11 curves of non-squarefree level which can be found in the list above. Remark. Using the Riemann-Hurwitz formula, we can compute the genus of any Atkin-Lehner quotient from the information on the following table. More- over, we can compute as well the number of elliptic cycles, nW , corresponding to the Atkin-Lehner quotient by W ￿= {id} and their orders as follows. Let us denote m(W ) = {m : ωm ∈W} and r(W ) such that 2r(W ) = #W . 2r(W )−1nW = (e￿2 + e ￿ 3)/2 + ￿ m∈m(W ) ￿ n∈Lm n, where, for k = 2, 3, e￿k = ek if k /∈ m(W ) and 0 otherwise. Moreover, all these elliptic cycles are of order two except for the following ones: e2 2r(W )−1 of order 4, if 2 ∈ m(W ), e3 2r(W ) of order 3, if 3 /∈ m(W ), e3 2r(W )−1 of order 6, if 3 ∈ m(W ). Proposition 4.2. The maximal Atkin-Lehner quotient of all the curves in the list of Proposition 4.1 has a rational CM point. In particular, they are all isomorphic to P1Q. Proof. Since the maximal Atkin-Lehner quotients of all the curves in the list are genus 0 curves, we only need to prove that all of them do have a rational CM point. All the curves in the list but 15 have a rational elliptic cycle. For these 15 values of (D,N) there exists a rational CM point, for example the CM 4.2 A detailed case: the curve X(22, 3) 135 point by the order Λ of discriminant D(Λ) given in the following table: (D,N) D(Λ) (6, 47) −52 (14, 23) −11 (39, 1) −7 (65, 1) −7 (111, 1) −19 (D,N) D(Λ) (6, 71) −52 (15, 11) −7 (39, 4) −7 (95, 1) −7 (143, 2) −20 (D,N) D(Λ) (14, 15) −11 (21, 19) −15 (62, 7) −20 (95, 3) −120 (194, 1) −20 4.2 A detailed case: the curve X(22, 3) In this section we are going to deal with the explicit uniformization of the Shimura curve X(22, 3). It follows from [Ogg83] that it is an hyperelliptic curve of genus 3, but it can be seen as well from the table in Proposition 4.1, since the quotient by ω66 has genus 0. At the same time we will study as well all its Atkin-Lehner quotients and in order to do so we will start by its maximal quotient, X(22, 3)+. We are going to use that both X(22, 1)+ and X(22, 3)+ are curves isomor- phic to P1Q, even though X(22, 3)W22 has genus 1. In the previous chapter we computed a Hauptmodul defined over Q for X(22, 1)+, t+22. Together with this function we computed the corresponding kroneckerian polynomial of level 3, Ψ3(X,Y ) = ΨΓ(22,1)+,3,t+22(X,Y ), which provided a singular model for the curve X(22, 3)W22 , and allowed us to obtain as well a singular model for X(22, 3)+, cf. Corollary 2.3. Now, we are going to make use of both this kroneckerian poly- nomial and the explicit equation for X(22, 3)+. Recall that the Hauptmodul t+22 was determined by the values at (three of) the elliptic cycles: P t+22(P ) CM(Z[i]) 1 CM(Z[ 1+ √−3 2 ]) 27 16 CM(Z[ 1+ √−11 2 ]) ∞ CM(Z[ √−22]) 0 136 Uniformization of some Shimura curves with full Atkin-Lehner quotient of genus 0 First of all, we need to properly choose a Hauptmodul for X(22, 3)+, t+22,3, and to do so we will take three rational CM points in X(22, 3)+. Since we are interested in relating this function to t+22, it will be convenient to choose, whenever possible, three CM points which are preimages of the elliptic cycles of X(22, 1)+. In order to decide on the rationality of these points in X(22, 3)+, and since the polynomial Ψ3(X,Y ) is known, the following lemma will be useful. Lemma 4.2. Consider the curve X(D,M)W . Assume that it is isomorphic to the projective line over Q and let f be a Hauptmodul over Q. Let N = pr for an odd prime p not dividing DM . Let P ∈ X(D,MN)W (C) be such that its image in X(D,M)W is elliptic and consider the natural birrational map X(D,MN)W → X(D,M)W ×X(D,M)W τ ￿→ (τ,ωNτ). Then, the image of the point P is rational in X(D,MN)￿W,ωN ￿ if any of the following conditions holds: (1) [Q(f(P )) : Q] ≤ 2, Gal(Q(f(P ))|Q) = ￿σ￿, f(P ) = σ(f(ωNP )) and (f(P ), f(ωNP )) is not a singularity of ΨΓ,N,f (X,Y ). (2) f(P ) = f(ωNP ) ∈ Q ∪ {∞}. Remark. In the hypotheses of the proposition above, Theorems 2.2 and 2.3 yield that the number of branches through (f(P ), f(ωNP )) is at most 2, and provides a description of the singularities. Since in this case we explicitly know the polynomial ΨΓ,N,f (X,Y ), we can compute the number of branches through (f(P ), f(ωNP )) as eN (P ) e1(P ) ordf(ωNP )ΨΓ,N,f (f(P ), X) = e1(ωNP ) e1(P ) ordf(ωNP )ΨΓ,N,f (f(P ), X). Proof. In the first case, the result is clear. The only case which needs to be considered is the second one when (P,ωNP ) is a singularity of ΨΓ,N,f (X,Y ). In this situation, Theorem 2.3 gives us that there exist two non-tangent branches through (P,ωNP ) and therefore there exist a couple of points P ￿, P ￿￿ ∈ X(D,MN)W (C) corresponding to this singularity. Moreover P ￿ + P ￿￿ is a rational divisor and therefore it is enough to prove that these points are not fixed by ωN . 4.2 A detailed case: the curve X(22, 3) 137 The elliptic points in X(D,M)W can be of two different types, the ones coming from elliptic points in X(D,M) and the fixed points by elements in W . Let us begin by the latter ones, that is, assume that the image of P in X(D,M)W is fixed by an element ωm ∈W , m > 1, m￿DM . Then, Theorem 4.1 yields that it must come from a CM point by an order of discriminant −m or −4m in X(D,M). However, if P is fixed by ωN it must come from a CM point by an order of discriminant −N or −4N in X(D,MN) and therefore its image in X(D,M) will be obtained from a CM point by an order containing this one and therefore of discriminant −N ￿ or −4N ￿ for some N ￿ such that N/N ￿ is a square. Since gcd(m,N) = 1, this is clearly a contradiction. Notice that it was not necessary in the previous case, but the condition f(P ) = f(ωNP ) yields that the order corresponding to P is the same both in X(D,M)W and X(D,MN)W . Keeping this in mind, we will now focus on the case that the image of P in X(D,M)W (and in X(D,MN)W ) is a CM point by Z[i] or Z[ 1+ √−3 2 ]. In order for it to be fixed by ωN , we must have N = 2 or N = 3, respectively. The first case cannot happen since we are assuming that N is odd. Therefore, we only have to deal with the case N = 3. Assume that (P,ωNP ) corresponds to a singular point. In this case we reach as well a contradiction, since Z[ 1+ √−3 2 ] is a principal ideal domain and the singularity provides a primitive element of norm 9 in this ring, which does not exist. Let us call π+22,3 : X(22, 3)W22 → X(22, 1)+ the natural projection, which is a morphism of curves ramified, at most, at the preimages of the elliptic cycles of X(22, 1)+. Since the kroneckerian polynomial gives a model for X(22, 3)W22 inside P1Q × P1Q and the corresponding morphism to X(22, 1)+ with the model provided by t+22 is given by the projection on the first component, we can easily study the ramification of π+22,3. Denote by Pi the CM points in X(22, 1)+ corresponding to orders of discriminant −i, for i = 3, 4, 11, 88, which correspond to the elliptic cycles in this curve. Recall that the values of the Hauptmodul t+22 at these points are 2716 , 1,∞, 0 respectively. For any of these values, t+22(Pi), the factorization of Ψ3(f(Pi), X) is as follows: i Ψ3(t + 22(Pi), X) 3 16(−27 + 16X)(−75 + 64X)3 4 (−49 + 16X)4 11 65536(−49 + 48X)2 88 ￿ 7425− 17568X + 12544X2￿2 138 Uniformization of some Shimura curves with full Atkin-Lehner quotient of genus 0 Thus, we obtain a set of rational points on the curve Ψ3(X,Y ) = 0. Recall that the elliptic order of the point Pi in the curve X(22, 1)+ is given by e1(Pi) =  2, i = 11, 22, 3, i = 3, 4, i = 4. Then using the above remark we get that each of these points corresponds to a certain number of points on the curve X(22, 3)W22 as stated in the following table: (P,ω3P ) mult(P,ω3P )Ψ3 ( 2716 , 27 16 ) 1 ( 2716 , 75 64 ) 1 (1, 4916 ) 1 (∞,∞) 2 (∞, 4948 ) 1 In particular, by Lemma 4.2, all of them define rational points on the curve X(22, 3)+. Now, since this is a curve of genus 0 with rational points we can define a Hauptmodul t+22,3 over Q by assigning three different rational values at 3 of the above points. Let us fix, once and for all, t+22,3 such that P t+22,3(P ) ( 2716 , 75 64 ) 1 (1, 4916 ) 0 (∞,∞) ∞ Lemma 4.3. Assume that X(D,M)W and X(D,MN)￿W,ωN ￿ have both genus 0 and let f, g be Hauptmoduln for these curves over a field K. Then for a, b ∈ K, a ￿= b, there exist R1(g), R2(g) ∈ K(g) such that (f − a)(f ◦ ωN − a) = R1(g), (f − b)(f ◦ ωN − b) = R2(g). Moreover, R1(g)(f − b)−R2(g)(f − a) + (a− b)(f − a)(f − b) = 0. 4.2 A detailed case: the curve X(22, 3) 139 Proof. It is only necessary to observe that (f − a)(f ◦ ωN − a) is invariant by ￿Γ(D,MN),W ￿. Remark. It is worth noting that the functions Ri(g) can be written as a quo- tient of two coprime polynomials of degree at most ψ(N) with both functions sharing the same denominator. The last equation in the lemma allows us to recover the values of f from those of g. In the case of the pole of g, if it corresponds to a point such that it is both a pole of f and f ◦ ωN , then since R1(g)R2(g) = f−af−b f◦ωN−af◦ωN−b , lim g→∞ R1(g) R2(g) = 1. Otherwise, if f takes a finite value f0 at a point corresponding to a pole of g but f ◦ ωN has a pole or the other way around, lim g→∞ R1(g) R2(g) = f0 − a f0 − b . Finally, if both f and f ◦ωN take finite values f0 and f ￿0 at a point corresponding to a pole of g, we have that lim g→∞ R1(g) R2(g) = f0 − a f0 − b f ￿0 − a f ￿0 − b . Back to our case, we consider the function (t+22 − 1)(t+22 ◦ ω3 − 1) ∈ Q(t+22,3) and comparing its zeros and poles with those of t+22,3, we get R1(t + 22,3) = (t + 22 − 1)(t+22 ◦ ω3 − 1) = a0 (t+22,3) 4 (t+22,3 − b0)2 for certain a0, b0 ∈ Q∗. Similarly, R2(t + 22,3) = (t + 22 − 2716 )(t+22 ◦ ω3 − 2716 ) = a1 (t+22,3 − 1)3(t+22,3 − a2) (t+22,3 − b0)2 for certain a1, a2 ∈ Q∗. Moreover, by the remark above, a0 = a1. Lemma 4.4. Let K be a number field and f a Hauptmodul for X(D,M)W defined over K. Then, for any N > 1 coprime to DM and any a, b ∈ K, a ￿= b, the polynomial Ψ ￿ X − Y 2(b− a) + b+ a 2 , bX − aY b− a + ab ￿ 140 Uniformization of some Shimura curves with full Atkin-Lehner quotient of genus 0 where Ψ(X,Y ) = ΨΓ,N,f (X + ￿ X2 − Y ,X − ￿ X2 − Y ) has ((f − a)(f ◦ ωN − a), (f − b)(f ◦ ωN − b)) as a root and provides a model for X(D,MN)￿W,ωN ￿ over K. Proof. It follows from Corollary 2.3, after writing f + f ◦ ωN 2 = (f − a)(f ◦ ωN − a)− (f − b)(f ◦ ωN − b) 2(b− a) + b+ a 2￿ f − f ◦ ωN 2 ￿2 = ￿ f + f ◦ ωN 2 ￿2 − f · f ◦ ωN f · f ◦ ωN = b(f − a)(f ◦ ωN − a)− a(f − b)(f ◦ ωN − b) b− a + ab. Now, using this lemma, we can explicitly compute a polynomial P ∈ Q[X,Y ] such that P ((t+22 − 1)(t+22 ◦ ω3 − 1), (t+22 − 2716 )(t+22 ◦ ω3 − 2716 )) = 0. Imposing that P (R1(t + 22,3), R2(t + 22,3)) = 0 we obtain a unique solution for a0, a2, b0 ∈ Q, namely, (R1(X), R2(X)) = ￿ 121X4 64(3X + 1)2 , 121(X − 1)3(2X + 1) 128(3X + 1)2 ￿ . The unicity of the solution could be predicted thanks to the following well- known result on the unicity up to linear rational transformations of proper parametrizations of rational irreducible algebraic curves, cf. [SWPD08]. Recall that, as usual, the degree of a rational function is the maximum of the degrees of the numerator and the denominator, after common factors are removed. Theorem 4.2. Consider an algebraic irreducible affine plane curve C of genus 0 over C with equation Ψ(X,Y ) = 0, with Ψ(X,Y ) ∈ C[X,Y ] irreducible. Then, there exist Q1(t), Q2(t) ∈ C(t) with degQ1 = degY Ψ, degQ2 = degX Ψ such that Ψ(Q1(t), Q2(t)) = 0. Moreover, Q1 and Q2 with these conditions are unique up to homographic transformation of t. 4.2 A detailed case: the curve X(22, 3) 141 According to this theorem, R1(X) and R2(X) are unique up to homographic transformations of the variable X, but no such transformation maps R1 and R2 to rational fractions of the same shape, in particular, we obtain that the solution had to be unique. Now, using the equation given in Lemma 4.3 together with Proposition 3.7, we can easily compute the automorphic derivative of t+22,3 from that of t + 22. Proposition 4.3. Da(t+22,3) +R(t + 22,3) = 0, where R(t) = 1452t6 − 2684t5 + 4327t4 + 1300t3 + 5046t2 − 2880t− 81 (2t− 3)2(2t+ 1)2(11t2 − 2t+ 3)2 . Observe that the zeros of the denominator of R are the values of t+22,3 at the elliptic points in X(22, 3)+, and thus we obtain the values at the CM points by Z[ 1+ √−3 2 ], Z[ 1+ √−11 2 ], Z[ √−33], Z[√−66], namely − 12 ,∞, 32 and the zeros of 11X2−2X+3 respectively. However as many values as desired can be computed as long as we know the value of t+22 at the corresponding point, for example we know that t+22 takes the value 516 at the CM point by Z[ √−5]. Then, since Ψ3( 5 16 , X) = 64(−5 + 16X)2 ￿ 6241− 10381X + 4624X2￿ , we can choose the point corresponding to ( 516 , 5 16 ), which is a CM point by Z[ √−5] in X(22, 3)+, the value of t+22,3 can be obtained solving the system (( 516 − 1)2, ( 516 − 2716 )2) = ￿ 121X4 64(3X + 1)2 , 121(X − 1)3(2X + 1) 128(3X + 1)2 ￿ which gives the value −1. Now, we are interested in computing uniformizing functions giving the canon- ical model and the corresponding equations for all the Atkin-Lehner quotients of X(22, 3). We will illustrate the procedure with a couple of cases and we will sum up the information for the whole set of 16 curves. In order to do so, we begin by studying the genus of the different curves and the ramification of the natural projections between them. Since the ramification points of the projec- tion coincide with the fixed points of the involutions we are adding, which are controlled by Theorem 4.1, and its number can be deduced from the number of such CM points in X(22, 3), which is given by Theorem 1.7, we can easily 142 Uniformization of some Shimura curves with full Atkin-Lehner quotient of genus 0 obtain the genus from the genus of X(22, 3), which is 3, using Riemann-Hurwitz formula. In particular, among the 16 curves we are considering, 6 have genus 0, in- cluding X(22, 3)+, which we have already studied. We will begin by the 5 genus 0 curves remaining and will compute a Hauptmodul over Q for those having one, and an equation as a conic for those who do not. Proposition 4.4. The curves X(22, 3)W , W ￿ W66, having genus 0 are those with W ∈ {￿ω66￿, ￿ω2,ω33￿, ￿ω3,ω22￿, ￿ω6,ω11￿, ￿ω3,ω11￿}. (1) X(22, 3)￿ω3,ω11￿ does not have rational (or even real) points. Its field of rational functions is Q ￿ t+22,3, ￿ −11(t+22,3)2 + 2t+22,3 − 3 ￿ and an equation for this curve over Q is Y 2 + 11X2 − 2X + 3 = 0. (2) The rest of these curves have rational points. There exist functions Ui such that Q(X(22, 3)￿ωi,ω66￿) = Q(Ui), i = 1, 2, 3, 6, defined as follows: i U2i −Da(Ui) 6 2t+22,3+1 2t+22,3−3 648−285U26−12044U46+69794U66+26796U86+8019U106 (−1+U6)2U26 (1+U6)2(27+50U26+99U46 ) 2 3 −2t+22,3 + 3 4(50787−64284U23+29266U43−5852U63+451U83 ) (−2+U3)2(2+U3)2(99−62U23+11U43 ) 2 2 −2t+22,3 − 1 − 4(−2592+771U22+9908U42+6690U62+1716U82+275U102 ) U22 (4+U22 ) 2(27+26U22+11U42 ) 2 1 1−U61+U6 32(11−132U21+388U41−516U61+579U81−516U101 +388U121 −132U141 +11U161 ) (−1+U1)2(1+U1)2(11−18U21+41U41−18U61+11U81 ) 2 Moreover, we can rewrite U1 = U3 − U2 2 . The action of the Atkin-Lehner involution ωd on such a Hauptmodul U is given by a rational linear function Rd(X) ∈ Q(X) such that U ◦ωd = Rd(U) 4.2 A detailed case: the curve X(22, 3) 143 as described in the following table: R2 R3 R6 R11 R22 R33 R66 U6 −X −X X X −X −X X U3 −X X −X −X X −X X U2 X −X −X −X −X X X U1 − 1X 1X −X −X 1X − 1X X Proof. This proof runs along the same lines we followed when uniformizing the curves of level N = 1 in the previous chapter. In the case of X(22, 3)￿ω3,ω11￿, the map to X(22, 3)+ ramifies at the CM points by Z[ √−66]. Since we know the values of the function t+22,3 at these points, namely the zeros of 11X2 − 2X + 3, we can take a function￿ k(11(t+22,3) 2 − 2t+22,3 + 3) for a certain integral and squarefree value of k for which it is defined over Q. In order to find the value of k we will evaluate this function at certain CM points to obtain conditions on k. For example, for the CM points by Z[ √−5], t+22,3 takes the value −1, so the function above takes the value 4√k which must lie in the Hilbert Class Field of Q( √−5) which is Q(√5, i). Similarly, if we evaluate the function ￿ k(11(t+22,3) 2−2t+22,3+3) t+22,3 at the CM points by Z[ 1+ √−11 2 ] it takes the value √ 11k which must lie in Q( √−11), since this field has class number 1. Therefore we conclude that k = −1 and the result follows. For the curve X(22, 3)￿ω6,ω66￿, its projection to X(22, 3)+ ramifies at the CM points by Z[ √−33], where t+22,3 is 32 , and by Z[ 1+ √−3 2 ], where t + 22,3 is − 12 . Thefore, we can take U6 = ￿ k 2t+22,3+1 2t+22,3−3 . Now, if we evaluate this function at the CM point by the ring of integers of Q( √−11), t+22,3 is ∞, hence we obtain that √ k ∈ Q(√−11). Similarly, for the CM points by Z[√−5], we obtain￿ k 5 ∈ Q( √ 5, i). Therefore k = 1. For the curve X(22, 3)￿ω3,ω66￿, the projection to X(22, 3)+ ramifies at the CM points by Z[ 1+ √−11 2 ] and Z[ √−33], so we can consider U3 = ￿ k(2t+22,3 − 3). Evaluating this function at the CM points by the ring of integers of Q( √−5) 144 Uniformization of some Shimura curves with full Atkin-Lehner quotient of genus 0 and Q( √−3), we obtain k = −1. Now, for the curve X(22, 3)￿ω2,ω66￿, we can take U2 = U3U6. For the curve X(22, 3)￿ω66￿, we consider the projection to X(22, 3)￿ω6,ω66￿, which ramifies at the CM points by the ring of integers of Q( √−11), where U6 takes the values ±1. Therefore, we can take U1 = ￿ kU6−1U6+1 . Now evaluating at the CM points by the rings of integers of Q( √−3) and Q(√−5), where U6 takes the values − 12 ,± √ 5 5 respectively, we obtain that k = −1. The automorphic derivatives are obtained using Proposition 3.7. The equations for all the remaining curves can be computed using these functions. Let us show how to obtain an hyperelliptic model for X(22, 3). Consider the projection to X(22, 3)￿ω66￿ which ramifies at the 8 CM points by Z[ √−66]. The values of U1 at these points are the roots of the polynomial 11X8 − 18X6 + 41X4 − 18X2 + 11. Therefore, Q(X(22, 3)) = Q ￿ U1, ￿ k(11U81 + 18U 6 1 + 41U 4 1 − 18U21 + 11) ￿ . To find out the value of k, we evaluate the function at the CM points by the rings of integers of Q( √−3) and Q(√−11), where U1 takes the values ±1 and 0,∞, respectively. In this way we find that k = −1. Similarly, we can work out all the remaining curves. For some of them it is useful to use the CM points by the order Z[3i], where t+22,3 takes the value 0 and the corresponding Ring Class Field is Q(i, √−3). We collect in the following proposition a set of generators providing an el- liptic or hyperelliptic model for all the curves X(22, 3)W which are not of genus 0. Proposition 4.5. (1) For W ∈ {￿ω2,ω11￿, ￿ω2,ω3￿, ￿ω6,ω22￿}, X(22, 3)W are elliptic curves. For each of them, the polynomial PW (X) in the following table provides a system of generators of the field of rational functions Q(X(22, 3)W ) = Q ￿ t+22,3, ￿ PW (t + 22,3) ￿ and the corresponding equation for the curve Y 2 = PW (X). 4.2 A detailed case: the curve X(22, 3) 145 W PW (X) ￿ω2,ω11￿ −(2X + 1)(2X − 3)(11X2 − 2X + 3) ￿ω2,ω3￿ (2X − 3)(11X2 − 2X + 3) ￿ω6,ω22￿ (2X + 1)(11X2 − 2X + 3) (2) The curves X(22, 3)￿ωi￿, for i = 3, 11, 33, are genus 1 curves without real points. For each of them, the polynomial Pi(X) and the function fi in the following table provide a system of generators of the field of rational functions Q(X(22, 3)￿ωi￿) = Q ￿ fi, ￿ Pi(fi) ￿ and the corresponding equation for the curve, Y 2 = Pi(X). i fi Pi(X) 3 U3 −(11X4 − 62X2 + 99) 11 U6 −(99X4 + 50X2 + 27) 33 U2 −(11X4 + 26X2 + 27) (3) The curves X(22, 3)￿ωi￿, for i = 2, 6, 22, are genus 2 curves. For each of them, the polynomial Pi(X) and the function fi in the following table provide a system of generators of the field of rational functions Q(X(22, 3)￿ωi￿) = Q ￿ fi, ￿ Pi(fi) ￿ and the corresponding equation for the curve Y 2 = Pi(X). i fi Pi(X) 2 U2 −(X2 + 4)(11X4 + 26X2 + 27) 6 U6 (X2 − 1)(99X4 + 50X2 + 27) 22 U3 −(X2 − 4)(11X4 − 62X2 + 99) 146 Uniformization of some Shimura curves with full Atkin-Lehner quotient of genus 0 (4) The curve X(22, 3) is an hyperelliptic curve of genus 3. Its field of rational functions is Q(X(22, 3)) = Q ￿ U1, ￿ P (U1) ￿ and the corresponding equation for the curve is Y 2 = −(11X8 − 18X6 + 41X4 − 18X2 + 11). 4.3 Other cases Once we have studied the uniformization of the curve X(22, 3) in detail, other curves X(D,N) where a Hauptmodul for X(D, 1)+ is known and such that X(D,N)+ has genus 0 can be studied in the same way. We will compilate in this section some more cases. Since we have focused on discriminants D = 6, 10, 22 we will restrict to these three discriminants. Moreover, we will study the three smallest values of N in each of these three cases, including the case (22, 3) above. Some of these cases for D = 6, 10 have already been studied before, namely (6, 5), (6, 7), (10, 3), (10, 7). The three first ones can be found in [Elk98] and the last one in [Yan13], [Tu14]. As far as we know, Hauptmoduln for the cases (6, 11) and (10, 9) have not been obtained before. We will use the tools we have already seen to recover or obtain Hauptmoduln for all those quotients of X(D,N) of genus 0 over Q. In some cases the approach above can be simplified by relating the Hauptmodul to the Hauptmodul of some genus 0 intermediate curves, which happens for all the curves studied in [Elk98] and [Tu14]. We will illustrate how to deal with these cases with the curve X(6, 5). Another situation where we can simplify the computations is that of composite levels, we will deal with this situation for the curve X(10, 9). Sometimes, we will obtain genus 0 curves which are given as a conic. In this case, if such a curve has a rational point, then it is indeed the projective line over Q, by means, for example, of the natural parametrization by lines through one rational point. We resume in the following lemma the equations involved, which allow us to build the Hauptmodul from the component functions and to recover the component functions from the Hauptmodul. Lemma 4.5. Given a conic in the projective plane of the form C : X2 + αY 2 − βZ2 = 0 4.3 Other cases 147 with a point [a : b : 1] over a field K, then it admits the following proper parametrization over K, f : P1K → C, with f(t) = [−at2 − 2αbt+ αa : bt2 − 2at− αb : t2 + α] which has inverse f−1([x : y : 1]) = x− a y − b . However, in some cases it will be more convenient to use other even more obvious parametrizations, for example, for the conics of the form XY = α or (X − Y )(X + Y ) = α. 4.3.1 The curve X(22, 5) This case is very similar to the model case we studied above. On the one hand, all the curves appearing of genus bigger than 2 are hyperelliptic. On the other hand, the computation of the Hauptmodul t+22,5 for X(22, 5)+ involves only slightly harder computations. The uniformization of X(22, 5)+ is given in the following proposition. Proposition 4.6. There exists a Hauptmodul for the curve X(22, 5)+, t+22,5, defined by the following values (t+22, t + 22 ◦ ω5) t+22,5 (1, 1) 0 (1, 729784 ) 1 (∞,∞) ∞ and it satisfies (t+22 − 1)(t+22 ◦ ω5 − 1) = 112253(t+22,5 − 1)4(t+22,5)2 256(125(t+22,5) 2 − 175t+22,5 + 64)2 , (t+22 − 2716 )(t+22 ◦ ω5 − 2716 ) = 121(25(t+22,5) 2 − 35t+22,5 + 16)3 256(125(t+22,5) 2 − 175t+22,5 + 64)2 . 148 Uniformization of some Shimura curves with full Atkin-Lehner quotient of genus 0 Moreover, Da(t+22,5) +R(t + 22,5) = 0, where R(X) = P (X) 4X2(5X − 4)2(1375X3 − 3650X2 + 3295X − 1024)2 and P (X) =5(28359375X8 − 162250000X7 + 421556250X6 − 669567500X5 + 731069375X4 − 564580300X3 + 294187120X2 − 91344896X + 12582912). In the following table we give the values of t+22,5 at CM(Λ) by certain orders Λ: Λ t+22,5 Z[i] 0 Z[ 1+ √−11 2 ] ∞ Z[ √−5] 45 Z[ √−110] α : 1375α3 − 3650α2 + 3295α− 1024 = 0 Z[5i] 1 Z[ 1+5 √−3 2 ] 1±√−15 10 Next we will give Hauptmoduln for the four remaining genus 0 curves, since in this case all of them have rational points. Proposition 4.7. There exist functions Ui = Ui,110 such that Q(X(22, 5)￿ωi,ω110￿) = Q(Ui), i = 1, 2, 5, 10, defined as follows: i U2i 10 5t+22,5 5t+22,5−4 5 −5t+22,5 2 −5t+22,5 + 4 4.3 Other cases 149 and U1 = U5 U2 + 2 . The action of the Atkin-Lehner involution ωd on such a Hauptmodul U is given by a rational linear function Rd(X) ∈ Q(X) such that U ◦ ωd = Rd(U) as described in the following table: R2 R5 R10 R11 R22 R55 R110 U10 −X −X X X −X −X X U5 −X X −X −X X −X X U2 X −X −X −X −X X X U1 −X − 1X 1X 1X − 1X −X X Remark. The expressions for the automorphic derivatives can be computed from that of t+22,5 as in the model case using Proposition 3.7. We do not include them now because they are considerably larger. Finally, models for the rest of the curves in this case are given in the following proposition. Proposition 4.8. (1) For W ∈ {￿ω5,ω11￿, ￿ω2,ω5￿, ￿ω2,ω11￿}, X(22, 5)W are elliptic curves. For each of them, the polynomial PW (X) in the following table provides a system of generators of the field of rational functions Q(X(22, 5)W ) = Q ￿ t+22,5, ￿ PW (t + 22,5) ￿ and the corresponding equation for the curve Y 2 = PW (X). W PW (X) ￿ω5,ω11￿ −5X(1375X3 − 3650X2 + 3295X − 1024) ￿ω2,ω5￿ 1375X3 − 3650X2 + 3295X − 1024 ￿ω2,ω11￿ (−5X + 4)(1375X3 − 3650X2 + 3295X − 1024) 150 Uniformization of some Shimura curves with full Atkin-Lehner quotient of genus 0 (2) The curves X(22, 5)W , forW ∈ {￿ω10,ω22￿, ￿ωi￿}i=2,5,11, are genus 2 curves. For each of them, the polynomials PW (X) and the function fW in the fol- lowing table provide a system of generators of the field of rational functions Q(X(22, 5)W ) = Q ￿ fW , ￿ PW (fW ) ￿ and the corresponding equation for the curve Y 2 = PW (X). W fW PW (X) ￿ω10,ω22￿ t+22,5 5X(5X − 4)(1375X3 − 3650X2 + 3295X − 1024) ￿ω2￿ U2 −(11X6 + 14X4 + 19X2 + 20) ￿ω5￿ U5 −(11X6 + 146X4 + 659X2 + 1024) ￿ω11￿ U10 5X6 − 34X4 + 109X2 + 256) (3) The curves X(22, 5)￿ωi￿, for i = 10, 22, 55, are genus 3 hyperelliptic curves. For each of them, the polynomial Pi(X) and the function fi in the following table provide a system of generators of the field of rational functions Q(X(22, 5)￿ωi￿) = Q ￿ fi, ￿ Pi(fi) ￿ and the corresponding equation for the curve Y 2 = Pi(X). i fi Pi(X) 10 U10 −(X2 − 1)(5X6 − 34X4 + 109X2 − 256) 22 U5 −(X2 + 4)(11X6 + 146X4 + 659X2 + 1024) 55 U2 −(X2 − 4)(11X6 + 14X4 + 19X2 + 20) (4) The curve X(22, 5) is an hyperelliptic curve of genus 5. Its field of rational functions is Q(X(22, 5)) = Q ￿ U1, ￿ P (U1) ￿ and the corresponding equation for the curve is Y 2 = −(64X12 + 275X10 + 660X8 + 818X6 + 660X4 + 275X2 + 64). 4.3 Other cases 151 4.3.2 The curve X(22, 7) The case of level 7, even though quite similar from a theoretical point of view to the cases we have already seen, presents already some more computational difficulties, in part due to the higher degree of the projection but also to the fact that most of the ramification points are no longer rational. However it is still possible to follow a similar approach to obtain a Hauptmodul for X(22, 7)+. Proposition 4.9. There exists a Hauptmodul for the curve X(22, 7)+, t+22,7, defined by the following values (t+22, t + 22 ◦ ω7) t+22,7 ( 2716 , 27 16 ) 0 ( 7564 , 75 64 ) 1 ( 516 , 5 16 ) ∞ and it satisfies (t+22 − 1)(t+22 ◦ ω7 − 1) = 121(18(t+22,7) 2 − 18t+22,7 + 1)4 256(324(t+22,7) 4 − 486(t+22,7)3 + 171(t+22,7)2 − 6t+22,7 + 1)2 , (t+22 − 2716 )(t+22 ◦ ω7 − 2716 ) = 1089(t+22,7) 2(36(t+22,7) 2 − 42t+22,7 + 7)3 256(324(t+22,7) 4 − 486(t+22,7)3 + 171(t+22,7)2 − 6t+22,7 + 1)2 . Moreover, Da(t+22,7) +R(t + 22,7) = 0, where R(X) = P (X) X2(36X2 − 36X + 1)2(45X2 − 48X + 4)2(48X2 − 56X + 9)2 and P (X) =4(1310940288X10 − 5781798144X9 + 10240292160X8 − 9287887680X7 + 4594990032X6 − 1260832608X5 + 201956472X4 − 18590520X3 + 953761X2 − 23680X + 288). The values of t+22,7 at the CM points by certain orders Λ are given in the following 152 Uniformization of some Shimura curves with full Atkin-Lehner quotient of genus 0 table: Λ t+22,7 Z[ 1+ √−3 2 ] 0 Z[ √−5] ∞ Z[ 1+3 √−3 2 ] 1 Z[ √−14] α : 36α2 − 36α+ 1 = 0 Z[ √−77] α : 45α2 − 48α+ 4 = 0 Z[ √−154] α : 48α2 − 56α+ 9 = 0 Proposition 4.10. (1) For W ∈ {￿ω2,ω77￿, ￿ω11,ω14￿, ￿ω14,ω22￿}, the curves X(22, 7)W are genus 0 curves. The first two of them have rational points and the third one does not. There exist functions Ui = Ui,154 such that Q(X(22, 7)￿ωi,ω154￿) = Q(Ui), i = 2, 11, defined as follows: i Ui 2 1+ ￿ 36(t+22,7) 2−36t+22,7+1 3t+22,7 11 2+ ￿ 45(t+22,7) 2−48t+22,7+4 3t+22,7 The action of the Atkin-Lehner involution ωd on such a Hauptmodul U is given by a rational linear function Rd(X) ∈ Q(X) such that U ◦ωd = Rd(U) as described in the following table: R2 R7 R11 R14 R22 R77 R154 U2 X 4−6X 6−X 4−6X 6−X 4−6X 6−X 4−6X 6−X X X U11 −5+4X −4+X −5+4X −4+X X X −5+4X −4+X −5+4X −4+X X On the other hand, Q(X(22, 7)￿ω14,ω22￿) = Q ￿ 12t+22,7 − 7, ￿ −3(48(t+22,7)2 − 56t+22,7 + 9) ￿ , 4.3 Other cases 153 and the corresponding equation is X2 + Y 2 − 22 = 0. (2) The curves X(22, 7)W for W in the following table are hyperelliptic curves. For each of them, the polynomial PW (X) provides a system of generators of the field of rational functions Q(X(22, 7)W ) = Q ￿ UW , ￿ PW (UW ) ￿ and a corresponding equation for the curve Y 2 = PW (X). W UW PW (X) ￿ω2,ω7￿ t+22,7 −3(45X2 − 48X + 4)(48X2 − 56X + 9) ￿ω7,ω11￿ t+22,7 −3(36X2 − 36X + 1)(48X2 − 56X + 9) ￿ω7,ω22￿ t+22,7 (45X2 − 48X + 4)(36X2 − 36X + 1) ￿ω14￿ U11 −(27X4 − 224X3 + 882X2 − 928X + 291) ￿ω77￿ U2 −(27X4 − 112X3 + 520X − 320X + 48) ￿ω154￿ U2 X4 − 8X3 + 45X2 − 28X + 4 ￿ω2,ω11￿ t+22,7 −3(36X2 − 36X + 1)(45X2 − 48X + 4)(48X2 − 56X + 9) ￿ω2￿ U2 −(X4 − 8X3 + 45X2 − 28X + 4)(27X4 − 112X3 + 520X2 − 320X + 48) ￿ω11￿ U11 −(X4 − 48X3 + 246X2 − 272X + 89)(27X4 − 224X3 + 882X2 − 928X + 291) (3) The remaining curves X(22, 7)W are non-hyperelliptic curves. (a) The natural map of X(22, 7)￿ω22￿ to X(22, 7)￿ω7,ω22￿ × X(22,7)+ X(22, 7)￿ω14,ω22￿ provides a model for this curve in P3 with affine equations Y 2 = (45X2 − 48X + 4)(36X2 − 36X + 1), Z2 = 22−X2. (b) The natural map of X(22, 7) to X(22, 7)￿ω2,ω77￿ × X(22,7)+ X(22, 7)￿ω11,ω14￿ × X(22,7)+ X(22, 7)￿ω14,ω22￿ 154 Uniformization of some Shimura curves with full Atkin-Lehner quotient of genus 0 provides a model for this curve in P4 as the intersection of the 3 hy- perquadrics Y 2 = (36X2 − 36X + 1), Z2 = (45X2 − 48X + 4), T 2 = −3(48X2 − 56X + 9). (c) The natural map of X(22, 7)￿ω7￿ to X(22, 7)￿ω2,ω7￿ × X(22,7)+ X(22, 7)￿ω7,ω22￿ provides a model for this curve in P3 with affine equations Y 2 = −3(45X2 − 48X + 4)(48X2 − 56X + 9), Z2 = (45X2 − 48X + 4)(36X2 − 36X + 1). The birrational map to P2 such that [X : Y : Z : 1] ￿→ [Y : Z : 2(48X2 − 56X + 9)] = [U : V : 1] has image the non-singular planar quartic curve 7 + 18Y 2 + 7Y 4 − 6Z2 − 38Y 2Z2 − 49Z4 = 0. Remark. Some of the genus 1 curves above have some obvious rational points which would allow us to compute a Weierstrass model for them, for example, we have seen that X(22, 7)￿ω7,ω22￿ has equation Y 2 = (45X2 − 48X + 4)(36X2 − 36X + 1) and (0, 2) is an obvious rational point. A Weierstrass model for it is Y 2 = X(X2 + 150X − 7) and the expression for the function corresponding to the variable X in this model, instead of t+22,7, is 8− 192t+ 189t2 + 4√4− 192t+ 1917t2 − 3348t3 + 1620t4 9t2 , where t = t+22,7. Note that even though the equation is somewhat simpler, the corresponding functions are fairly more involved. Therefore, since we are both interested in the functions and the corresponding equations, we will restrict to the models given above. 4.3 Other cases 155 4.3.3 The curve X(6, 5) This case is easier than the ones above, since X(6, 5)￿ω2,ω3￿ has genus 0, and therefore there exists a morphism X(6, 5)￿ω2,ω3￿ → X(6, 1)￿ω2,ω3￿ = X(6, 1)+ between curves of genus 0. For discriminantD = 6 this happens forN = 5, 7, 13. In this situation, we can break the computation of a Hauptmodul forX(6, 5)+ into two steps. First we follow the approach found in [Elk98] to compute directly a Hauptmodul for the curve X(6, 5)￿ω2,ω3￿. In order to do so, we can make use of the knowledge of the polynomial Ψ5(X,Y ) = ΨΓ(6,1)+,5,t+6 (X,Y ) to simplify the discussion of cases. Secondly, we can obtain a Hauptmodul for X(6, 5)+ together with some additional properties from this Hauptmodul. Even though the case N = 5 is quite straightforward, the same approach can be used to solve as well the cases D = 6, N = 7 and D = 10, N = 3, 7. We will begin by choosing a Hauptmodul defined over Q for the curve X(6, 5)￿ω2,ω3￿, we will simply call it f , since we will later normalize it and rename it, as follows: (t+6 , t + 6 ◦ ω5) f (1, 4218505211771561 ) 0 ( 4218505211771561 , 1) 1 (∞, 152881138240 ) ∞ It can be checked as we did before, that all the points we chose to define the Hauptmodul are non-singular on the curve. Therefore, f is a Hauptmodul defined over Q. Consequently, t+6 − 1 = af4(f2 + bf + c), a, b, c ∈ Q, and f2+bf+c vanishes at the couple of points where t+6 (P ) = t + 6 ◦ω5(P ) = 1, or equivalently at the elliptic points by Z[i]. Moreover, since f ◦ω5 is a Hauptmodul and ω5 is involutive, there exist x, y, z ∈ Q such that f ◦ ω5 = xf + y zf − x and by the choice of the values above, we obtain that f ◦ ω5 = f−1df−1 for some d ∈ Q. Therefore, t+6 ◦ ω5 − 1 = a￿(f − 1)4(f2 + bf + c) (df − 1)6 . 156 Uniformization of some Shimura curves with full Atkin-Lehner quotient of genus 0 Observe that the quadratic polynomial in the numerator is the same in both expressions since its corresponding divisor is fixed by ω5. In particular, if we evaluate both equations at suitable points where we know the values of f , one of the expressions will become an identity, and the other will yield an equation relating these parameters: • evaluating at f = 0: 4218505211771561 − 1 = a￿c, • evaluating at f =∞: 152881138240 − 1 = a ￿ d6 , • evaluating at f = 1: 4218505211771561 − 1 = a(1 + b+ c), • evaluating at f = 1/d: 152881138240 − 1 = a(cd 2+db+1) d6 . If we solve these equations, we obtain that we can write all the parameters in terms of d, namely, a − 14641d6138240(−1+d) b − 58071715430400+25937424601d525937424601d5 c 5807171543040025937424601d6 a￿ 14641d 6 138240 Then, we can either substitute the resulting expressions for t+6 and t + 6 ◦ ω5 into Ψ5(t+6 , t + 6 ◦ ω5) = 0 or use that the above expression of t+6 in terms of f must contain a quadratic square factor. Either way, we obtain a unique possible value for d, namely, d = − 504121 . In particular, t+6 = ￿ 605− 2772t+ 5292t2￿ ￿3025 + 6930t+ 10584t2￿2 5536128125 and f ◦ ω5 = 121(1− f) 121 + 504f . This allows us to compute the value of f at any point where we know the values 4.3 Other cases 157 of t+6 and t + 6 ◦ ω5. In particular, we obtain the following values: D(Λ) (t+6 , t + 6 ◦ ω5) f −4 (1, 1) α : 7056α2 + 5544α+ 3025 = 0 −24 (0, 0) α : 5292α2 − 2772α+ 605 = 0 −40 ( 2312125 , 2312125 ) − 1114 −120 ( 57763375 , 57763375 ) 1136 In addition, we obtain a Hauptmodul for X(6, 5)+, for example, we can take F = f + f ◦ω5 = 121+505f2121+504f . We could also have taken f · f ◦ω5 = − 121504 (F − 1). Now, consider t+6,5 the Hauptmodul for X(6, 5)+ defined by the following three values: D(Λ) t+6,5 −4 ∞ −24 1 −40 −1 Comparing the values of F and t+6,5 at these three points we obtain that t+6,5 = 7F + 33 2(14F + 11) = 4 ￿ 605 + 2079f + 441f2 ￿ 3025 + 5544f + 7056f2 , and in particular we obtain that the value of t+6,5, for example, at the CM point by Z[ √−30] is 6164 . Now, since the morphism X(6, 5)￿ω2,ω3￿ → X(6, 5)+ ramifies at the CM points by Z[ √−10] and Z[√−30], we can take a Hauptmodul of X(6, 5)￿ω2,ω3￿ such that U22,3 = k t+6,5 − 6164 t+6,5 + 1 for a certain squarefree integer k. Then, substituting t+6,5 in this expression in terms of f , yields that k = −1 and we can take U2,3 = ￿ − t + 6,5 − 6164 t+6,5 + 1 = 7 24 36f − 1 14f + 11 . 158 Uniformization of some Shimura curves with full Atkin-Lehner quotient of genus 0 Moreover, for this Hauptmodul the action of ω5 is U2,3 ◦ ω5 = −U2,3 and the expression of t+6 in terms of U2,3 is t+6 = ￿ 3 + 128U22,3 ￿ ￿ 228 + 117U2,3 + 272U22,3 ￿2 125(−3 + 4U2,3)6 . Then, using Proposition 3.7 we can obtain the automorphic derivative of U2,3 in terms of that of t+6 and from that of U2,3, using the same result, we can obtain the automorphic derivative of t+6,5. In the following propositions we give these results together with Hauptmoduln for all the genus 0 curves and generating functions and equations for the others. Proposition 4.11. There exists a Hauptmodul for the curve X(6, 5)+, t+6,5, defined by the following values D(Λ) t+6,5 −4 ∞ −24 1 −40 −1 and it satisfies t+6 · t+6 ◦ ω5 = (−1 + t+6,5)2 ￿ 919− 173t+6,5 + 64(t+6,5)2 ￿2 (−1 + 4t+6,5)6 , (t+6 − 1)(t+6 ◦ ω5 − 1) = 729(4− 5t+6,5)4 (1− 4t+6,5)6 . Moreover, Da(t+6,5) +R(t + 6,5) = 0, where R(X) = 3 ￿ 5120X4 − 6928X3 + 12841X2 − 24304X + 13307￿ 4(X − 1)2(X + 1)2(64X − 61)2 . The values of t+6,5 at the CM points by certain orders Λ are given in the following table: 4.3 Other cases 159 Λ t+6,5 Z[ √−30] 6164 Z[5i] 45 Z[ 1+5 √−3 2 ] 1 4 Next we will give Hauptmoduln for all the remaining genus 0 curves with rational points. Proposition 4.12. There exist functions UK = U (6,5) K such that Q(X(6, 5)￿ωk:k∈K￿) = Q(UK) for K ∈ {{2, 3}, {2, 5}, {6, 10}, {2, 15}, {3, 10}, {5, 6}, {6}, {30}}. They can be chosen as follows K U2K {2, 3} − t + 6,5− 6164 t+6,5+1 {2, 5} − t + 6,5− 6164 t+6,5−1 {6, 10} −(t+6,5 − 6164 ) {2, 15} t + 6,5+1 t+6,5−1 {3, 10} t+6,5 − 1 {5, 6} t+6,5 + 1 and U6 = U6,10 − 118 U5,6 − 14 , U30 = U5,6 + U3,10. The action of the Atkin-Lehner involution ωd on such a Hauptmodul U is given by a rational linear function Rd(X) ∈ Q(X) such that U ◦ ωd = Rd(U) as 160 Uniformization of some Shimura curves with full Atkin-Lehner quotient of genus 0 described in the following table: R2 R3 R5 R6 R10 R15 R30 U2,3 X X −X X −X −X −X U2,5 X −X X −X −X X −X U6,10 −X −X −X X X X −X U2,15 X −X −X −X −X X X U3,10 −X X −X −X X −X X U5,6 −X −X X X −X −X X U6 − 1X 2+11X−11+2X 11−2X2+11X 11−2X2+11X 2+11X−11+2X − 1X X U30 −X − 2X 2X 2X − 2X −X X Moreover, for a suitable determination of the root we have that t+6 = 1− 27(7 + 24U2,3)4 ￿ 1 + U22,3 ￿ 125(−3 + 4U2,3)6 . Remark. The expressions for the automorphic derivatives can be computed from that of t+6,5 as in the previous cases using Proposition 3.7. Finally, models for the rest of the curves in this case are given in the following proposition. Proposition 4.13. (1) For i = 2, 10, X(6, 5)￿ωi￿ are conics without real points. We have Q(X(6, 5)￿ω2￿) = Q(U2,15, U2,5) Q(X(6, 5)￿ω10￿) = Q(U6,10, U3,10) and the corresponding equations are 3X2 + 128Y 2 + 125 = 0, X2 + Y 2 + 3 64 = 0, respectively. 4.3 Other cases 161 (2) The curves X(6, 5)W , for W ⊂ ￿ω3,ω5￿, are genus 1 curves. For each of them, functions fW , gW providing a system of generators of the function field Q(X(6, 5)W ) and a polynomial PW giving the corresponding equation for the curve f2W = PW (gW ) are displayed in the following table: W fW , gW PW (X) ￿ω3,ω5￿ U2,3U3,10(t+6,5 + 1), t+6,5 −(X − 1)(X + 1)(X − 6164 ) ￿ω3￿ U2,3(U23,10 + 2), U3,10 −(X2 + 2)(X2 + 364 ) ￿ω5￿ U2,5(U25,6 − 2), U5,6 −(X2 − 12564 )(X2 − 2) ￿ω15￿ U2,15(U26,10 + 364 ), U6,10 (X2 − 12564 )(X2 + 364 ) ￿1￿ 8U6,10U30, U30 −16X4 + 61X2 − 64 4.3.4 The curve X(6, 7) This case is, apart from some increase in the computational complexity, analo- gous to the previous one. Proposition 4.14. There exists a Hauptmodul for the curve X(6, 7)+, t+6,7, defined by the following values D(Λ) t+6,7 −3 ∞ −24 0 −84 1 and it satisfies t+6 · t+6 ◦ ω7 = (t+6,7) 2 ￿−2016 + 3129t+6,7 − 518(t+6,7)2 + 81(t+6,7)3￿2 16(−4 + 7t+6,7)6 , (t+6 − 1)(t+6 ◦ ω7 − 1) = ￿ 16 + 3t+6,7 + 9(t + 6,7) 2 ￿4 16(4− 7t+6,7)6 . 162 Uniformization of some Shimura curves with full Atkin-Lehner quotient of genus 0 Moreover, Da(t+6,7) +R(t + 6,7) = 0, where R(X) = 3 ￿ 1024− 5152X + 13777X2 − 15753X3 + 8505X4￿ 4(−1 +X)2X2(−32 + 81X)2 . The values of t+6,7 at the CM points by certain orders Λ are given in the following table: Λ t+6,7 Z[ √−42] 3281 Z[ 1+7 √−3 2 ] 4 7 Z[ √−10] 13 Z[ 1+ √−19 2 ] 4 3 Next we will give Hauptmoduln for all the remaining genus 0 curves with rational points. Proposition 4.15. There exist functions UK = U (6,7) K such that Q(X(6, 7)￿ωk:k∈K￿) = Q(UK) for K ∈ {{2, 3}, {2, 21}, {3, 7}, {3, 14}, {6, 7}, {6, 14}, {42}}. They can be chosen as follows K U2K {2, 3} −3 t + 6,7−1 t+6,7− 3281 {2, 21} 3t+6,7 {3, 7} −3 t + 6,7 t+6,7− 3281 {3, 14} t + 6,7 t+6,7−1 {6, 7} 3(t+6,7 − 1) {6, 14} −(t+6,7 − 3281 ) 4.3 Other cases 163 and U42 = U2,21 + U6,7. The action of the Atkin-Lehner involution ωd on such a Hauptmodul U is given by a rational linear function Rd(X) ∈ Q(X) such that U ◦ ωd = Rd(U) as described in the following table: R2 R3 R6 R7 R14 R21 R42 U2,3 X X X −X −X −X −X U2,21 X −X −X −X −X X X U3,7 −X X −X X −X X −X U3,14 −X X −X −X X −X X U6,7 −X −X X X −X −X X U6,14 −X −X X −X X X −X U42 3 X −X − 3X − 3X −X 3X X Moreover, for a suitable determination of the root we have that t+6 = 1− 49 ￿ 162 + 27U2,3 + 44U22,3 ￿4 4(−27 + 10U2,3)6 ￿ 3 + U22,3 ￿ . Remark. The expressions for the automorphic derivatives can be computed from that of t+6,7 as in the previous cases using Proposition 3.7. Finally, models for the rest of the curves in this case are given in the following proposition. Proposition 4.16. (1) For i = 3, 6, 21, X(6, 7)￿ωi￿ are conics without rational points. We have Q(X(6, 7)￿ω3￿) = Q(U2,3, U3,7) Q(X(6, 7)￿ω6￿) = Q(U6,7, U6,14) Q(X(6, 7)￿ω21￿) = Q(U2,21, U6,14) 164 Uniformization of some Shimura curves with full Atkin-Lehner quotient of genus 0 and the corresponding equations are, respectively, 32X2 + 49Y 2 + 3 · 81 = 0, X2 + 3Y 2 + 49 27 = 0, X2 + 3Y 2 − 32 27 = 0. (2) The curves X(6, 7)W , for W ⊂ ￿ω2,ω7￿, are genus 1 curves. For each of them, functions fW , gW providing a system of generators of the function field Q(X(6, 7)W ) and a polynomial PW giving the corresponding equation for the curve f2W = PW (gW ) are displayed in the following table: W fW , gW PW (X) ￿ω2,ω7￿ U2,3U2,21(t+6,7 − 3281 )/3, t+6,7 −X(X − 1)(X − 3281 ) ￿ω2￿ U2,3(U22,21 − 3227 ), U2,21 −3(X2 − 3)(X2 − 3227 ) ￿ω7￿ U3,7(U26,7 + 4927 ), U6,7 −3(X2 + 3)(X2 + 4927 ) ￿ω14￿ U3,14(U26,14 + 4981 ), U6,14 (X2 − 3281 )(X2 + 4981 ) {id} 18U6,14U42, U42 −27X4 − 34X2 − 243 4.3.5 The curve X(6, 11) Now, we are going to deal with the curve X(6, 11). From a theoretical point of view it is similar to the previous cases, however the determination of a Haupt- modul is considerably harder from a computational point of view. Proposition 4.17. There exists a Hauptmodul for the curve X(6, 11)+, t+6,11, defined by the following values D(Λ) t+6,11 −24 0 −88 ∞ −132 1 4.3 Other cases 165 and it satisfies t+6 · t+6 ◦ ω11 = A(t+6,11)￿ 3 + 22t+6,11 + 275(t + 6,11) 2 ￿6 (t+6 − 1)(t+6 ◦ ω11 − 1) = 729 ￿−1− 41t+6,11 − 751(t+6,11)2 + 441(t+6,11)3￿4￿ 3 + 22t+6,11 + 275(t + 6,11) 2 ￿6 , where A(X) = 16X2 ￿−891− 36036X − 367202X2 −231132X3 − 9164859X4 + 3886472X5￿2 . Moreover, Da(t+6,11) +R(t + 6,11) = 0, where R(X) = 3 ￿ 9 + 507X + 13531X2 + 285618X3 + 2261179X4 − 2224101X5 + 1771561X6￿ 4(−1 +X)2X2 (3 + 118X + 1331X2)2 . In the following table we give the values of t+6,11 at CM(Λ) by certain orders Λ: Λ t+6,11 Z[ 1+11 √−3 2 ] α : 3 + 22α+ 275α 2 = 0 Z[ √−66] α : 3 + 118α+ 1331α2 = 0 Z[ √−10] − 19 Λ t+6,11 Z[ 1+ √−19 2 ] −1 Z[ √−30] 35 Z[ 1+ √−43 2 ] − 149 In this case the computations are harder and therefore, since the solution has rational coefficients, it is useful to find p-adic approximations. Then we can check that the approximated value up to a certain precision is indeed a solution. Finally we can conclude that it is the solution we are looking for thanks to Theorem 4.2. In this case, we worked with 101-adic approximations. Next we will give Hauptmoduln for all the remaining genus 0 curves with rational points. Proposition 4.18. There exist functions UK = U (6,11) K such that Q(X(6, 11)￿ωk:k∈K￿) = Q(UK) 166 Uniformization of some Shimura curves with full Atkin-Lehner quotient of genus 0 for K ∈ {{2, 33}, {3, 22}, {6, 11}, {66}}. They can be chosen as follows K U2K {2, 33} −t+6,11 {3, 22} 2 t + 6,11 t+6,11−1 {6, 11} −2(t+6,11 − 1) and U66 = U2,33 − 1 U6,11 − 2 . The action of the Atkin-Lehner involution ωd on such a Hauptmodul U is given by a rational linear function Rd(X) ∈ Q(X) such that U ◦ ωd = Rd(U) as described in the following table: R2 R3 R6 R11 R22 R33 R66 U2,33 X −X −X −X −X X X U3,22 −X X −X −X X −X X U6,11 −X −X X X −X −X X U66 −1+X −1+2X 1 2X 2X−1 2(X−1) 2X−1 2(X−1) 1 2X −1+X −1+2X X Remark. The expressions for the automorphic derivatives can be computed from that of t+6,11 as in the previous cases using Proposition 3.7. Finally, models for the rest of the curves in this case are given in the following proposition. Proposition 4.19. (1) X(6, 11)￿ω6,ω22￿ is a conic without real points. We have Q(X(6, 11)￿ω6,ω22￿) = Q ￿ t+6,11, ￿ −(1331(t+6,11)2 + 118t+6,11 + 3) ￿ and the corresponding equation is 1331X2 + 118X + Y 2 + 3 = 0. 4.3 Other cases 167 (2) ForW ∈ {￿ω2,ω3￿, ￿ω2,ω11￿, ￿ω3,ω11￿, ￿ω6￿, ￿ω22￿, ￿ω33￿}, the corresponding curves X(6, 11)W are genus 1 curves. For each of them, functions fW , gW providing a system of generators of the function field Q(X(6, 11)W ) and a polynomial PW giving the corresponding equation for the curve f2W = PW (gW ) are displayed in the following table: W fW , gW PW (X) ￿ω2,ω3￿ U6,11 ￿ −(1331(t+6,11)2 + 118t+6,11 + 3), t+6,11 2(X − 1)(3 + 118X + 1331X2) ￿ω2,ω11￿ U2,33U6,11 ￿ −(1331(t+6,11)2 + 118t+6,11 + 3), t+6,11 −2X(X − 1)(3 + 118X + 1331X2) ￿ω3,ω11￿ U2,33 ￿ −(1331(t+6,11)2 + 118t+6,11 + 3), t+6,11 X(3 + 118X + 1331X2) ￿ω6￿ 2 ￿ −(1331(t+6,11)2 + 118t+6,11 + 3), U6,11 −5808 + 5560X2 − 1331X4 ￿ω22￿ (U23,22 − 2) ￿ −(1331(t+6,11)2 + 118t+6,11 + 3), U3,22 −4 ￿ 3− 62X2 + 363X4￿ ￿ω33￿ ￿ −(1331(t+6,11)2 + 118t+6,11 + 3), U2,33 −3 + 118X2 − 1331X4 (3) The curves X(6, 11)W , for W ∈ {￿ω2￿, ￿ω3￿, ￿ω11￿}, are genus 2 curves. For each of them, functions fW , gW providing a system of generators of the function field Q(X(6, 11)W ) and a polynomial PW giving the corresponding equation for the curve f2W = PW (gW ) are displayed in the following table: W fW , gW PW (X) ￿ω2￿ U6,11 ￿ −(1331(t+6,11)2 + 118t+6,11 + 3), U2,33 −2 ￿ 1 +X2 ￿ ￿ 3− 118X2 + 1131X4￿ ￿ω3￿ 4 ￿ −(1331(t+6,11)2+118t+6,11+3) U36,11 , U3,22 ￿−2 +X2￿ ￿3− 62X2 + 313X4￿ ￿ω11￿ 4U2,33 ￿ −(1331(t+6,11)2 + 118t+6,11 + 3), U6,11 −2 ￿−2 +X2￿ ￿5008− 4760X2 + 1131X4￿ (4) The curve X(6, 11) is an hyperelliptic curve of genus 3. Its function field is generated by the functions Q(X(6, 11)) = Q ￿ (−1 + 2U26 )2 8 ￿ −(1331(t+6,11)2 + 118t+6,11 + 3), U6 ￿ 168 Uniformization of some Shimura curves with full Atkin-Lehner quotient of genus 0 and the corresponding equation is Y 2 = −(19− 318X + 2133X2 − 7350X3 + 13943X4 − 14700X5 + 8532X6 − 2544X7 + 304X8). 4.3.6 The curve X(10, 3) Finally, we will move on to the discriminant 10 case. Therefore, we will use the Hauptmodul t+10 for the curve X(10, 1)+ introduced in Theorem 2.4 and further studied in Theorem 3.1. Since the map X(10, 3)W10 → X(10, 1)+ has degree 4, the computations can be carried out without any special difficulties. Proposition 4.20. There exists a Hauptmodul for the curve X(10, 3)+, t+10,3, defined by the following values D(Λ) t+10,3 −3 0 −120 ∞ −20 1 and it satisfies t+10 · t+10 ◦ ω3 = 729 ￿ 1− 11t+10,3 + 64(t+10,3)2 ￿2 4(−1 + t+10,3)2(1 + 80t+10,3)2 (t+10 − 1)(t+10 ◦ ω3 − 1) = (5− 32t+10,3)2(5 + 49t+10,3)2 4(−1 + t+10,3)2(1 + 80t+10,3)2 . Moreover, Da(t+10,3) +R(t + 10,3) = 0, where R(X) = 875− 8475X + 44403X2 − 44768X3 + 27648X4 36(−1 +X)2X2(−5 + 32X)2 . The values of t+10,3 at the CM points by certain orders Λ are given in the following 4.3 Other cases 169 table: Λ t+10,3 Z[ √−2] 532 Z[ 1+3 √−3 2 ] − 54 Z[3 √−2] − 549 Z[3 √−5] − 180 Z[ 1+ √−35 2 ] 1 28 Next we will give Hauptmoduln for all the remaining genus 0 curves with rational points. Proposition 4.21. There exist functions UK = U (10,3) K such that Q(X(10, 3)￿ωk:k∈K￿) = Q(UK) for K = {{2, 3}, {2, 5}, {2, 15}, {3, 5}, {3, 10}, {5, 6}, {30}}. They can be chosen as follows: K U2K {2, 3} 3(t+10,3 − 1) {2, 5} −5t+10,3 {2, 15} 30(t+10,3 − 532 ) {3, 5} −15 t + 10,3−1 t+10,3 {3, 10} 10 t + 10,3− 532 t+10,3−1 {5, 6} −6 t + 10,3− 532 t+10,3 and U30 = U2,15 + 4U5,6. The action of the Atkin-Lehner involution ωd on such a Hauptmodul U is given by a rational linear function Rd(X) ∈ Q(X) such that U ◦ ωd = Rd(U) as 170 Uniformization of some Shimura curves with full Atkin-Lehner quotient of genus 0 described in the following table: R2 R3 R5 R6 R10 R15 R30 U2,3 X X −X X −X −X −X U2,5 X −X X −X X −X −X U2,15 X −X −X −X −X X X U3,5 −X X X −X −X X −X U3,10 −X X −X −X X −X X U5,6 −X −X X X −X −X X U30 81 X −X − 81X − 81X −X 81X X Moreover, for a suitable determination of the root we have that t+10 = 27 2 − 27U2,5(−5 + 2U2,5) 3 5(−1 + 4U2,5)2 ￿ 5 + U22,5 ￿ . Remark. The expressions for the automorphic derivatives can be computed from that of t+10,3 as in the previous cases using Proposition 3.7. Finally, models for the rest of the curves in this case are given in the following proposition. Proposition 4.22. (1) For i = 2, 3, 5, X(10, 3)￿ωi￿ are conics without rational points. We have Q(X(10, 3)￿ω2￿) = Q(U2,3, U2,5) Q(X(10, 3)￿ω3￿) = Q(U2,3, U3,5) Q(X(10, 3)￿ω5￿) = Q(U2,5, U3,5) and the corresponding equations are, respectively, 5X2 + 3Y 2 + 15 = 0, 10X2 − Y 2 + 5 · 81 16 = 0, 6X2 + Y 2 + 75 16 = 0. 4.3 Other cases 171 (2) The curves X(10, 3)W , for W ⊂ ￿ω6,ω10￿, are genus 1 curves. For each of them, functions fW , gW providing a system of generators of the function field Q(X(10, 3)W ) and a polynomial PW giving the corresponding equation for the curve f2W = PW (gW ) are displayed in the following table: W fW , gW PW (X) ￿ω6,ω10￿ U2,3U2,5U3,515 , t+10,3 −2X(X − 1)(X − 532 ) ￿ω6￿ U5,6(U22,3 + 3), U2,3 −6(X2 + 3)(X2 + 8132 ) ￿ω10￿ U3,10(U22,5 + 5), U2,5 10(X2 + 5)(X2 + 2532 ) ￿ω15￿ U2,15(U23,5 + 7516 ), U3,5 −15(X2 − 5·8116 )(X2 + 7516 ) {id} 30 U30U2,5 , U30 −3(X4 + 222X2 + 6561) 4.3.7 The curve X(10, 7) Next, we deal with the curve X(10, 7)+. Like in the cases D = 6, 22, the compu- tations become more involved as the degree of the mapX(10, 7)W10 → X(10, 1)+ increases, even though, they can still be carried out. Moreover, Theorem 4.2 provides a way to check the result without having to go back to the computa- tions. Proposition 4.23. There exists a Hauptmodul for the curve X(10, 7)+, t+10,7, defined by the following values D(Λ) t+10,7 −35 0 −280 ∞ −20 1 and it satisfies t+10 · t+10 ◦ ω7 = (−25 + 32t+10,7)2 ￿ 2000− 1016t+10,7 − 6859(t+10,7)2 + 6561(t+10,7)3 ￿2 4(−1 + t+10,7)2 ￿ 10000− 20200t+10,7 − 919(t+10,7)2 + 13520(t+10,7)3 ￿2 , 172 Uniformization of some Shimura curves with full Atkin-Lehner quotient of genus 0 (t+10− 27 2 )(t+10◦ω7− 27 2 ) = (−20 + 27t+10,7)2 ￿ 400− 835t+10,7 + 484(t+10,7)2 ￿3 (−1 + t+10,7)2 ￿ 10000− 20200t+10,7 − 919(t+10,7)2 + 13520(t+10,7)3 ￿2 . Moreover, Da(t+10,7) +R(t + 10,7) = 0, where R(X) ∈ Q(X) is 750000− 4135000X + 10178675X2 − 15135155X3 + 14698603X4 − 8589408X5 + 2239488X6 4(−1 +X)2X2(−20 + 27X)2(−25 + 32X)2 . In the following table we give the values of t+10,7 at CM(Λ) by certain orders Λ: Λ t+10,7 Z[ √−10] 2532 Z[ 1+ √−3 2 ] 20 27 Z[ √−52] 2013 Λ t+10,7 Z[3 √−5] 100121 Z[ 1+3 √−3 2 ] 5 12 Z[ 1+ √−115 2 ] 16 23 Now, we are going to provide Hauptmoduln for all the quotients of genus 0 with rational points. Proposition 4.24. There exist functions UK = U (10,7) K such that Q(X(10, 7)￿ωk:k∈K￿) = Q(UK) for K = {{2, 5}, {2, 35}, {5, 7}, {5, 14}, {7, 10}, {10, 14}, {70}}. They can be cho- sen as follows: K U2K {2, 5} −5t+10,7 {2, 35} 10 t + 10,7− 2532 t+10,7−1 {5, 7} 70(t+10,7 − 2532 ) {5, 14} −14 t + 10,7− 2532 t+10,7 {7, 10} −35 t + 10,7−1 t+10,7 {10, 14} 7(t+10,7 − 1) 4.3 Other cases 173 and U70 = U5,14 − 72 U7,10 − 7 . The action of the Atkin-Lehner involution ωd on such a Hauptmodul U is given by a rational linear function Rd(X) ∈ Q(X) such that U ◦ ωd = Rd(U) as described in the following table: R2 R5 R7 R10 R14 R35 R70 U2,5 X X −X X −X −X −X U2,35 X −X −X −X −X X X U5,7 −X X X −X −X X −X U5,14 −X X −X X −X −X X U7,10 −X −X X X −X −X X U10,14 −X −X −X X X X −X U70 5 16X −5+10X −10+16X −5+8X 8(−1+2X) −5+8X 8(−1+2X) −5+10X −10+16X 5 16X X Moreover, for a suitable determination of the root we have that t+10 = 27 2 − ￿ 100 + 15U2,5 + 22U22,5 ￿3 ￿ 100 + 27U22,5 ￿ 5 ￿ 5 + U22,5 ￿ ￿ 500 + 200U2,5 + 141U22,5 + 52U 3 2,5 ￿2 . Remark. The expressions for the automorphic derivatives can be computed from that of t+10,7 as in the previous cases using Proposition 3.7. Finally, models for the rest of the curves in this case are given in the following proposition. Proposition 4.25. 1. For i = 5, 10, 35, X(10, 7)￿ωi￿ are conics without ra- tional points. We have Q(X(10, 7)￿ω5￿) = Q(U2,5, U5,7) Q(X(10, 7)￿ω10￿) = Q(U2,5, U10,14) Q(X(10, 7)￿ω35￿) = Q(U5,7, U10,14) 174 Uniformization of some Shimura curves with full Atkin-Lehner quotient of genus 0 and the corresponding equations are 14X2+Y 2+ 7 · 125 16 = 0, 7X2+5Y 2+35 = 0, X2−10Y 2− 5 · 49 16 = 0, respectively. 2. The curves X(10, 7)W , for W ⊂ ￿ω2,ω7￿, are genus 1 curves. For each of them, functions fW , gW providing a system of generators of the function field Q(X(10, 7)W ) and a polynomial PW giving the corresponding equation for the curve f2W = PW (gW ) are displayed in the following table: W fW , gW PW (X) ￿ω2,ω7￿ U2,5U5,7U10,1435 , t+10,7 −2X(X − 1)(X − 2532 ) ￿ω2￿ U2,35(U22,5 + 5), U2,5 10(X2 + 5)(X2 + 12532 ) ￿ω7￿ U7,10(U25,7 + 7·12516 ), U5,7 −35(X2 + 7·12516 )(X2 − 5·4916 ) ￿ω14￿ U5,14(U210,14 + 7), U10,14 −14(X2 + 7)(X2 + 4932 ) {id} 5(16U270−5)2U2,5 , U70 −(75− 280X + 528X2 − 896X3 + 768X4) 4.3.8 The curve X(10, 9) In this case, since we have a composite level it is easier to consider the map X(10, 9)W10 → X(10, 3)+, or even better in this caseX(10, 9)W10 → X(10, 3)W10 , since we have already seen that X(10, 3)W10 also has genus 0. We have already computed a Hauptmodul for X(10, 3)W10 , namely U (10,3)2,5 . For simplicity, we will denote it by u. In order to carry out the computations explicitly, even though it will not matter in the final result, it is necessary to choose one of the two determinations of the root for u. We will choose the one 4.3 Other cases 175 such that D(Λ) u −3 0 −3 · 32 − 52 −20 · 32 − 14 Then, we take the kroneckerian polynomial of level 9, ΨΓ(10,1)+,9,t+10(X,Y ), and recall that ΨΓ(10,1)+,9,t+10(t + 10, X) ∈ Q(X(10, 1)+)[X] is the irreducible poly- nomial of t10,3 ◦ ω9 over Q(X(10, 1)+). In particular, when considering this polynomial over the field Q(X(10, 3)W10), it is divisible by the irreducible poly- nomial of t+10,3 ◦ω9 over Q(X(10, 3)W10), which is a factor of degree 3. However, it is more convenient to replace t+10 by u and then we obtain a certain polynomial which could be defined as the kroneckerian polynomial of level 9 corresponding to the Hauptmodul u of X(10, 3)W10 , that is, we find an irreducible polynomial Ψ(X,Y ) =− 625− 750X − 300X2 − 40X3 − 750Y + 3825XY − 1440X2Y + 1680X3Y − 300Y 2 − 1440XY 2 − 1980X2Y 2 + 1752X3Y 2 − 40Y 3 + 1680XY 3 + 1752X2Y 3 + 896X3Y 3 such that Ψ(u, u ◦ ω9) = 0. We could indeed define a kroneckerian polynomial in this setting which would match this one, but since this situation will not appear again, having this explicit polynomial together with the properties of ΨΓ(10,1)+,3,t+10(X,Y ), ΨΓ(10,1)+,9,t+10(X,Y ) will suffice. Then, we consider the Hautpmodul t+10,9 of X(10, 9)+ defined by the values D(Λ) (u(P ), u(ω9P )) t + 10,9(P ) −3 · 32 (0,− 52 ) 0 −3 · 32 (− 52 ,− 52 ) ∞ −20 · 32 (− 14 ,− 14 ) 1 176 Uniformization of some Shimura curves with full Atkin-Lehner quotient of genus 0 and then we can relate t+10,9 and u as follows: u · u ◦ ω9 =a0 (t+10,9) 3 (t+10,9) 3 + b(t+10,9) 2 + ct+10,9 + d , (u+ 5/2)(u ◦ ω9 + 5/2) =a1 t+10,9 (t+10,9) 3 + b(t+10,9) 2 + ct+10,9 + d . Now, evaluating at the points such that t+10,9(P ) =∞, 1, we obtain that a0 = 25 4 , a1 = 25 · 81 4 . Moreover, (t+10,9) 2 = 81 u · u ◦ ω9 (u+ 5/2)(u ◦ ω9 + 5/2) . In particular, the values of t+10,9 at the poles of u and u ◦ ω9 are obtained evaluating the above expression at u =∞ and the corresponding value of u◦ω9, which can be computed from Ψ as a root of 112X3+219X2+210X − 5. Thus, the corresponding value of t+10,9 is a root of either X3 + 9X2 + 54X − 54, X3 − 9X2 + 54X + 54. Evaluating any of the two equalities relating the functions at the point where t+10,9 takes the value 1, we obtain that the value of this polynomial at 1 must be 100 and therefore, u · u ◦ ω9 =25 4 (t+10,9) 3 (t+10,9) 3 − 9(t+10,9)2 + 54t+10,9 + 54 , (u+ 5/2)(u ◦ ω9 + 5/2) =25 · 81 4 t+10,9 (t+10,9) 3 − 9(t+10,9)2 + 54t+10,9 + 54 . Then, using that t+10 = 27 ￿ 5 + 5u+ 8u2 ￿2 10(−1 + 4u)2 (5 + u2) , we obtain that t+10 · t+10 ◦ ω9 = A(t+10,9) C(t+10,9) , (t+10 − 1) · (t+10 ◦ ω9 − 1) = B(t+10,9) C(t+10,9) , 4.3 Other cases 177 where A(X), B(X), C(X) ∈ Q[X] are the following: A(X) =9 ￿ 99144− 107892X + 22356X2 + 17064X3 − 2214X4 + 153X5 + 289X6￿2 , B(X) =625(3 +X)2 ￿ 6− 4X +X2￿2 ￿−540 + 594X + 90X2 + 25X3￿2 , C(X) =4(−6 +X)2 ￿6 + 2X +X2￿2 ￿594− 216X − 99X2 + 121X3￿2 . Finally we can recover the expression for the automorphic derivative of t+10,9 from that of u or from that of t+10,9, using Proposition 3.7. Proposition 4.26. There exists a Hauptmodul for the curve X(10, 9)+, t+10,9, defined by the following values D(Λ) (t+10(P ), t + 10(ω9P )) t + 10,9(P ) −3 · 32 ( 272 , 867242 ) 0 −3 · 32 ( 867242 , 867242 ) ∞ −20 · 32 ( 289120 , 289120 ) 1 and it satisfies t+10 · t+10 ◦ ω9 = A(t+10,9) C(t+10,9) , (t+10 − 1) · (t+10 ◦ ω9 − 1) = B(t+10,9) C(t+10,9) , where A(X), B(X), C(X) ∈ Q[X] are the following: A(X) =9 ￿ 99144− 107892X + 22356X2 + 17064X3 − 2214X4 + 153X5 + 289X6￿2 , B(X) =625(3 +X)2 ￿ 6− 4X +X2￿2 ￿−540 + 594X + 90X2 + 25X3￿2 , C(X) =4(−6 +X)2 ￿6 + 2X +X2￿2 ￿594− 216X − 99X2 + 121X3￿2 . Moreover, Da(t+10,9) +R(t + 10,9) = 0, where R(X) ∈ Q(X) is 4 ￿ 11664− 8208X + 9108X2 + 888X3 − 3024X4 + 148X5 + 253X6 − 38X7 + 9X8￿ (−6 +X)2(−1 +X)2(2 +X)2(3 +X)2 (6− 3X +X2)2 . 178 Uniformization of some Shimura curves with full Atkin-Lehner quotient of genus 0 The values of t+10,9 at the CM points by certain orders Λ are given in the following table: Λ t+10,9 Z[ √−2] −3 Z[ 1+ √−35 2 ] 3 2 Z[3 √−2] −2 Z[ √−5] 6 Z[3 √−10] α : α2 − 3α+ 6 = 0 Next, we move on to the genus 0 quotients. In this case, only two more quotients have genus 0 and both of them have rational points. Proposition 4.27. There exist functions UK = U (10,9) K such that Q(X(10, 9)￿ωk:k∈K￿) = Q(UK) for K = {{2, 45}, {5, 18}}. They can be chosen as follows: K U2K {2, 45} t + 10,9−6 t+10,9+2 {5, 18} t + 10,9−1 t+10,9+3 The action of the Atkin-Lehner involution ωd on such a Hauptmodul U is given by a rational linear function Rd(X) ∈ Q(X) such that U ◦ ωd = Rd(U) as described in the following table: R2 R5 R9 R10 R18 R45 R90 U2,45 X −X −X −X −X X X U5,18 −X X −X −X X −X X 4.3 Other cases 179 Remark. The expressions for the automorphic derivatives can be computed from that of t+10,9 as in the previous cases using Proposition 3.7. Finally, models for the elliptic and hyperelliptic quotients are given in the following proposition. Proposition 4.28. (1) The curves X(10, 9)W , for W ∈ {￿ω2,ω9￿, ￿ω2,ω5￿, ￿ω5,ω9￿, ￿ω9,ω10￿, ￿ω10,ω18￿, ￿ω90￿}, are genus 1 curves. For each of them, functions fW , gW providing a system of generators of the function field Q(X(10, 9)W ) and a polynomial PW giving the corresponding equation for the curve f2W = PW (gW ) are displayed in the following table: W fW , gW PW (X) ￿ω2,ω9￿ ￿ −3(t+10,9 − 1)(t+10,9 − 6)((t+10,9)2 − 3t+10,9 + 6), t+10,9 −3(X − 6)(X − 1)(X2 − 3X + 6) ￿ω2,ω5￿ ￿ −3(t+10,9 − 1)(t+10,9 + 2)((t+10,9)2 − 3t+10,9 + 6), t+10,9 −3(X − 6)(X + 2)(X2 − 3X + 6) ￿ω5,ω9￿ ￿ −3(t+10,9 + 2)(t+10,9 + 3)((t+10,9)2 − 3t+10,9 + 6), t+10,9 −3(X + 2)(X + 3)(X2 − 3X + 6) ￿ω9,ω10￿ ￿ (t+10,9 − 6)(t+10,9 − 1)(t+10,9 + 2)(t+10,9 + 3), t+10,9 (X − 6)(X − 1)(X + 2)(X + 3) ￿ω10,ω18￿ ￿ −3(t+10,9 − 6)(t+10,9 + 3)((t+10,9)2 − 3t+10,9 + 6), t+10,9 −3(X − 6)(X + 3)(X2 − 3X + 6) ￿ω90￿ U5,18(U22,45 − 9), U2,45 −(X + 3)(X − 3)(3X2 + 5) (2) The curves X(10, 9)W , forW ∈ {￿ω2￿, ￿ω5￿, ￿ω18￿, ￿ω45￿}, are genus 2 curves. For each of them, functions fW , gW providing a system of generators of the function field Q(X(10, 9)W ) and a polynomial PW giving the corresponding equation for the curve f2W = PW (gW ) are displayed in the following table: W fW , gW ￿ω2￿ (U 2 2,45−1)2 8 ￿ −3(t+10,9 − 1)(t+10,9 + 2)((t+10,9)2 − 3t+10,9 + 6), U2,45 ￿ω5￿ (U 2 5,18−1)2 4 ￿ −3(t+10,9 + 2)(t+10,9 + 3)((t+10,9)2 − 3t+10,9 + 6), U5,18 ￿ω18￿ (U 2 5,18−1)2 4 ￿ −3(t+10,9 − 6)(t+10,9 + 3)((t+10,9)2 − 3t+10,9 + 6), U5,18 ￿ω45￿ (U 2 2,45−1)2 8 ￿ −3(t+10,9 + 2)(t+10,9 + 3)((t+10,9)2 − 3t+10,9 + 6), U2,45 180 Uniformization of some Shimura curves with full Atkin-Lehner quotient of genus 0 W PW (X) ￿ω2￿ −3 ￿ 5 + 3X2 ￿ ￿ 3 + 3X2 + 2X4 ￿ ￿ω5￿ −3 ￿ 3 +X2 ￿ ￿ 1− 3X2 + 6X4￿ ￿ω18￿ −3 ￿−5 + 9X2￿ ￿1− 3X2 + 6X4￿ ￿ω45￿ 3(−3 +X)(3 +X) ￿ 3 + 3X2 + 2X4 ￿ Chapter 5 Automorphic form expansions: arithmetic properties In the classical theory of modular forms, the existence of cusps, and therefore of Fourier expansions, plays a central role when carrying out explicit compu- tations. In the case of Shimura curves attached to rational non-split indefinite quaternion algebras, the absence of cusps prevents the existence of Fourier ex- pansions. Alternative expansions around CM points, which provide in some sense a suitable substitute in the general case, have been introduced in [Bay02] and further studied in [BT07a] for the modular case and in [BT07b, BT08] for the case of the Shimura curve of discriminant 6. Explicit expansions for some functions around certain CM points have already been computed in these papers. Some properties of these expansions, including the algebraicity of the coefficients in the case D = 6, have been studied in [BT08] as well. Expan- sions around CM points from a more geometric point of view have also been considered in [Mor94, Mor95, Mor11]. In this chapter, we generalize those expansions considered in [BT07b] to provide expansions for quaternionic automorphic forms. Then we relate the coefficients of these expansions to the values of certain differential operators, closely related to those studied in [Shi75a, Shi79] for certain types of modular forms, and then we follow a similar approach to the one in these papers to 181 182 Automorphic form expansions: arithmetic properties obtain some arithmetic properties of the coefficients. More precisely, we prove, for any discriminant and level, that the field generated by the coefficients of the expansion corresponding to an automorphic form is closely related to its field of definition and that, when suitably normalized, Shimura reciprocity law extends to all its coefficients. In this way, we recover some of the results in [Shi79] and [BT08], formulated over precise number fields instead of over Q. Using the algebraicity results of the coefficients of these expansions we can deduce the rationality of the automorphic derivative of an automorphic function, which is a classical result proved by Ihara [Iha74] using a completely different approach. In the next chapter, we will study how to compute explicit q-expansions of cusp forms for the cases we have studied in the previous chapter, and we will provide some explicit examples. 5.1 Shimura-Maaß operators and coefficients In this section, we will recall the definition and some properties of the Shimura- Maaß differential operators introduced in [Shi75a], a slight modification of a differential operator already considered by Maaß in [Maa52]. These differen- tial operators, when applied to an automorphic form, yield (generically) non- holomorphic functions which behave as automorphic forms of higher weight. We will prove that the values obtained when evaluating these functions at a CM point P is related to the coefficients of the q-expansion of the original form around P . However, in order to preserve the holomorphy, we will replace every Shimura-Maaß differential operator with a family of holomorphic differential op- erators, one for every CM point, and we will show that, for those corresponding to a certain CM point P , their values when applied to an automorphic form and evaluated at P coincide with those of the Shimura-Maaß. Moreover, we will prove some properties of these operators, including a transformation law, play- ing the same role as the automorphic behaviour of the original Shimura-Maaß operators. Recall that in Section 3.3, we considered q-expansions of automorphic func- tions with respect to the following local parameter qP : Definition. For P ∈ H and κ ∈ C∗, we define the local parameter around P with local constant κ, q = qP,κ : H→ C, as q(z) = κ z − P z − P . 5.1 Shimura-Maaß operators and coefficients 183 Lemma 5.1. Let q = qP,κ be a local parameter with constant κ at P ∈ H and t a non-negative integer. Let also δ2t = δκ,P2t = q − κ κ(P − P ) ￿ (q − κ) d dq − 2t ￿ . (1) For every non-negative integer t and F = F (q) analytic in a neighborhood of 0, δ2t ￿￿ (q − κ)2 κ(P − P ) ￿t F ￿ = ￿ (q − κ)2 κ(P − P ) ￿t+1 dF dq . (2) Let F = ￿ n≥0 anq n be an analytic function in a neighborhood of q = 0. Then, δ2t+2s ◦ · · · ◦ δ2t+2 ◦ δ2t ￿￿ (q − κ)2 κ(P − P ) ￿t F ￿ = ￿ (q − κ)2 κ(P − P ) ￿t+s+1 ds+1F dqs+1 . In particular,￿ δ2t+2s ◦ · · · ◦ δ2t+2 ◦ δ2t ￿￿ (q − κ)2 κ(P − P ) ￿t F ￿￿ (P ) = κt+s+1(s+ 1)! (P − P )t+s+1 as+1. Proof. The first equality is a direct computation and the second one follows immediately from the first one. Given Γ the group of units of norm 1 of an Eichler order in a rational indefi- nite quaternion algebra, we recall some of the notions introduced in Section 1.3 concerning automorphic functions and forms. Ak(Γ)K denotes the set of weight k (meromorphic) automorphic forms defined over a field K ⊂ C, which is in fact an A0(Γ)K-vector space of dimension 1. We will denote by Sk(Γ)K the K-vector space of weight k cusp forms defined over K and S∗(Γ)K the graded K-algebra of cusp forms. If K = C, we will omit the subscript. In the case of a non-split indefinite quaternion algebra, since there are no cusps, all holomorphic automorphic forms are cusp forms. Now, consider the Shimura-Maaß operator δ˜2t = 1 2πi ￿ d dz + 2t z − z ￿ , where d dz = 1 2 ￿ d dx − i d dy ￿ , z = x+ iy, cf. [Shi75a]. Let us also consider f = 1 (2πi)t ￿ (q − κ)2 κ(P − P ) ￿t￿ n≥0 anq n ∈ A2t(Γ)K . 184 Automorphic form expansions: arithmetic properties Proposition 5.1. (δ˜2t+2s ◦ · · ·◦ δ˜2t)(f)(P ) = ￿ κ 2πi(P − P ) ￿t+s+1 (s+1)!as+1. Proof. Since dz dz = 0, given a smooth function h = h(z, z) and a point P ∈ H we have the equality δ˜th(P ) = 1 2πi ￿ d dz + t z − P ￿ h(z, P ). And the results follows from the previous lemma after observing that δt is obtained from d dz + t z − P after applying the change of variables q(z) = κ z − P z − P . Now let De2t,κ,P = δ κ,P 2t+2(e−1) ◦ · · · ◦ δκ,P2t and D˜e2t = δ˜2t+2(e−1) ◦ · · · ◦ δ˜2t. Finally, for e = 1 we will not usually write the superindex and if, in addition, t = 0, we will abbreviate D = D0 = D˜0 = 1 2πi d dz . Proposition 5.2. (1) Let f = 1 (2πi)t ￿ (q − κ)2 κ(P − P ) ￿t￿ n≥0 an qn n! be analytic in a neighborhood of z = P . Then, (De2t,κ,P f)(P ) = ae ￿ κ 2πi(P − P ) ￿t+e . (2) Dt+s,κ,P (fg) = Dt,κ,P (f)g + fDs,κ,P (g), for any pair of functions f, g an- alytic at z = P . (3) Dt+s,κ,P ((z − P )−tf) = (z − P )−tDs,κ,P (f), for any function f analytic at z = P . (4) Dt,κ,P ￿(f |tα) = (Dt,κ,P (f))|t+2α for any function f analytic at z = P and any α ∈ GL+2 (R) such that α(P ￿) = P . (5) For any number field K, D0 : A0(Γ)K → A2(Γ)K . 5.2 Arithmeticity principle 185 Proof. The first assertion has been proved in the previous lemma and the second one is easily checked using the definition. To prove (3), we apply (2) to obtain the equality Dt+s,κ,P ((z − P )−tf) = Dt,κ,P ((z − P )−t)f + (z − P )−tDs,κ,P (f) and observe that (z − P )−t = ￿ q − κ κ(P − P ) ￿t . Now, by definition of Dt,κ,P , we obtain that Dt,κ,P ￿￿ q − κ κ(P − P ) ￿t￿ = 0 and therefore the result. To prove (4), let f be as in (1), this is f(z) = ￿ dqP,κ(z) dz ￿t F (qP,κ(z)). Then, f |2tα = at ￿ dqP ￿,κ(z) dz ￿t F (aqP ￿,κ(z)) for a ∈ C∗ such that qP,κ(αz) = aqP ￿,κ(z). According to the previous lemma, we have D2t,κ,P ￿(f |2tα) = at ￿ dqP ￿,κ(z) dz ￿t+1 F (aqP ￿,κ(z)) dqP ￿,κ(z) = ￿ dqP,κ(αz) dz ￿t+1 F (qP,κ(αz)) dqP,κ(αz) = (D2t,κ,P f)|2t+2α. Finally, (5) follows immediately, since, by definition, Df ∈ A2(Γ)K for any f ∈ A0(Γ)K . 5.2 Arithmeticity principle In this section we will study the field of definition of the coefficients of suitably normalized expansions of automorphic forms around CM points and we will show that Shimura reciprocity law relates expansions of an automorphic form at CM points by the same quadratic order, in the same way it relates the values of functions. The outline of the proof is essentialy the one in [Shi75a], which deals with a similar problem in a different case. 186 Automorphic form expansions: arithmetic properties Let H be a rational indefinite quaternion algebra of discriminant D and let O = O(D,N) ⊂ H be an Eichler order of level N , gcd(D,N) = 1. Fix an embedding Φ : H ￿→M(2,R). Let also K|Q be a quadratic imaginary field and Λ ⊂ OK be an order such that there exists an optimal embedding φ : Λ ￿→ O. We denote by KΛ the Ring Class Field of K corresponding to the order Λ. Let ε ∈ Λ be such that Λ = Z[ε] and let P ∈ H be the common fixed point by Φ(φ(Λ)). Consider ρ = Φ(φ(ε)). Finally, given A = ￿ a b c d ￿ ∈ M(2,R) and z ∈ H, we define λ(A, z) = cz + d. Lemma 5.2. Given α ∈ O a primitive element of norm N ￿, gcd(N ￿, ND) = 1, then O ∩ α−1Oα is an Eichler order of level NN ￿. Proof. Recall that O is an Eichler order of level N ￿ if and only if for every prime p ∈ Z, Op = O ⊗ Zp is conjugated to ￿ Zp Zp N ￿Zp Zp ￿ if p|N ￿ and after H⊗Qp is identified with M(2,Qp), and is a maximal Zp-order otherwise. On the one hand, if p ￿ N ￿, then N ￿ ∈ Z∗p. Therefore, α ∈ O∗p, which gives that α−1Opα = Op. Hence, α−1Opα∩Op is a maximal Zp-order if p ￿ N and is conjugated to ￿ Zp Zp NZp Zp ￿ = ￿ Zp Zp NN ￿Zp Zp ￿ otherwise. On the other hand, if p | N ￿, then Op is maximal, since p ￿ N . Hence, α−1Opα∩Op is the intersection of two maximal orders and, therefore, an Eichler order. To compute its level, since α is primitive and Op is maximal, Op,1αOp,1 is the set of all primitive elements of norm N ￿. In particular, γ1 ￿ 0 −1 N ￿ 0 ￿ γ2 = α for γ1, γ2 ∈ Op,1. Now, a direct computation shows that α−1Opα ∩Op = γ−12 ￿ Zp Zp N ￿Zp Zp ￿ γ2 and, consequently, α−1Opα ∩Op ∼= ￿ Zp Zp N ￿Zp Zp ￿ = ￿ Zp Zp NN ￿Zp Zp ￿ . Remark. An analogous result also holds if we let the norm of α be divisible by primes p | D. 5.2 Arithmeticity principle 187 Lemma 5.3. Fix b ∈ Z￿ {0}. There exist an integer n and a finite set S ⊂ Z such that for every a ∈ Z￿ S satisfying gcd(det(a Id+bρ), n) = 1, then (det(a Id+bρ))−1/2 λ(a Id+bρ, P ) is not a root of unity. Proof. Firstly, we observe that λ(a Id+bρ, P ) ∈ K. More precisely, we have that λ(a Id+bρ, P ) is a root of the characteristic polynomial of a Id+bρ, a polynomial defining K. Thus, (det(a Id+bρ))−1/2 λ(a Id+bρ, P ) ∈ K((det(a Id+bρ))1/2) and it is an extension of Q of degree at most 4. Consequently, if (det(a Id+bρ))−1/2 λ(a Id+bρ, P ) is a root of unity, it is a root of unity lying in an extension of degree 4 and there are only a finite number of these. In particular, there exist m ∈ Z and an mth-root of unity ξm such that (det(a Id+bρ))−1/2 λ(a Id+bρ, P ) ∈ Q(ξm). Hence it holds that (det(a Id+bρ))1/2 ∈ K(ξm). Now, we observe that det(a Id+bρ) is not a square except for a finite number of a, as det(a Id+bρ) = NormK|Q(a+bε) = (a+b tr(ε) 2 ) 2+b2(NormK|Qε− tr(ε) 2 4 ) and b2(NormK|Qε− tr(ε) 2 4 ) is a non-zero rational number (it is the discriminant of K up to a non-zero square factor) independent of a. Let S be this set of integers. Let now a be an integer not in S. Then, (det(a Id+bρ))1/2 defines a quadratic subextension of K(ξm). Therefore, there exists n (e.g. the product of the dis- criminants of the quadratic subextensions of K(ξm)), such that if we have gcd(det(a Id+bρ), n) = 1, then (det(a Id+bρ))1/2 /∈ K(ξm) and, in particular, (det(a Id+bρ))−1/2 λ(a Id+bρ, P ) is not a root of unity. Lemma 5.4. For every positive integer N , there exists b ∈ Z ￿ {0} such that gcd(det(a Id+bρ), N) = 1 for infinitely many integers a. Proof. Choosing b properly, we can rewrite the set {a+bε}a as {a+b￿ √ DK}a for some integer b￿. Then, we need prove that there exist infinitely many a such that gcd(det(Φ(φ(a+b￿ √ DK))), N) = 1. Since det(Φ(φ(a+b￿ √ DK))) = a2−b￿2DK , it is enough to prove that for every integer t, there exist infinitely many a such that gcd(a2 + t,N) = 1. And this follows easily, assuming first that N is prime and then using the Chinese Remainer Theorem to obtain the general case. 188 Automorphic form expansions: arithmetic properties Proposition 5.3. Following the notations above, there exists α ∈ H∗ such that α(P ) = P , α−1Oα∩O is an Eichler order admitting φ as an optimal embedding for Λ and (det(α))−1/2 λ(α, P ) is not a root of unity. Proof. Follows immediately from the previous lemmas, taking α ∈ φ(Λ). Let L be a number field. Let also Γ be the image in PSL2(R) of Φ(O1). Given a,w ∈ Z, w > 0, we fix the representative [a]w ∈ {0, . . . , w − 1} of the residue class a (mod w). Theorem 5.1. Let f ∈ A2r(Γ)L and g ∈ A2r+2e(Γ)L with r, e non-negative integers. Let P ∈ H be a complex multiplication point by the order Λ ⊂ K and let w be its elliptic order (by Γ). Then, if f, g are holomorphic at P and ordP g = [−r − e]w, (g−1Def)(P ) ∈ LK(j(P )) = L ·KΛ. Remark. • (Def)(P ) = 0 if e+ r ￿≡ 0 (mod w). • If e + r ￿≡ 0 (mod w), then there exists g ∈ A2r+2e(Γ)L·KΛ such that g(P ) ￿= 0. • More generally, if f ∈ A2r(Γ)L·KΛ is holomorphic at P and e ≥ 0, then there exists g ∈ A2r+2e(Γ)L·KΛ such that ordP Def = ord g. Proof. Let n0 = ordP f ≥ 0. We will proceed by induction on e in two steps. We will begin by proving the cases 0 ≤ e ≤ n0, reducing them to the case e = 0, and then we will prove the cases e > n0 by reduction to those ones. - e = 0: We need to prove that (f/g)(P ) ∈ L ·KΛ, but this is obvious, since f/g ∈ A0(Γ)L·KΛ and it is holomorphic at P . When we rewrite this in terms of the coefficients of the q-expansions of f and g, we obtain that: if we consider f = Qr ￿ n≥n0 an qn n! , g = Q r ￿ n≥n0 bn qn n! with an0 , bn0 ￿= 0 and Q = (q−κ) 2 2πiκ(P−P ) , then an0/bn0 ∈ L ·KΛ. - 0 < e ≤ n0: We will reduce this situation to the case e = 0. With the notations for the expansions which we gave in the previous case, we have Def = Qr+e ￿ n≥n0 an qn−e (n− e)! . 5.2 Arithmeticity principle 189 Let g1 ∈ A2r+2e(Γ)L·KΛ be such that ordP g1 = ordP Def = n0 − e. Let t ∈ A0(Γ)L·KΛ be such that t(P ) = 0. Then, Dt/t ∈ A2(Γ)L·KΛ has a simple pole at P and this allows us to write g1 = g ￿ ordP tDt t ￿e g2, where g2 ∈ A0(Γ)L·KΛ is holomorphic and non-vanishing at P . Therefore, evaluating ((Def)/g1)(P ) = an0n0! g2(P )(n0−e)!bn0 which lies in K · L Λ, since an0/bn0 and g2(P ) do, as we wanted to prove. - e > n0: Let α ∈ H∗ be the element given by Proposition 5.3. Then, if we consider h = f |2rαf , its value at P is not a root of unity, more precisely, h(P ) = λ−2r−2n0 . Now, applying De2r to both sides of the equality f |2rα = fh yields (Def)|2r+2eα = (Def)h+ ￿ p Cp(D e−pf)(Dph), which we can rewrite as: (Def)|2r+2eα− (Def)h = e−1￿ p=1 Cp(D e−pf)(Dph) + fDeh. Note that ordP ((De−pf)Dph) ≥ [−r − e + p]w + [−p]w ≥ [−r − e]w and for every p ∈ {1, . . . , e} we choose gp ∈ A2p(Γ)L·KΛ with order at P equal to [−p]w, which is less or equal than ordP Dph. Therefore, we obtain that ordP (De−pf) ≥ [−r − e+ p]w ≥ [−r − e]w − [−p]w = ordP (gg−1p ). Putting everything together, it follows that (Def)|2r+2eα− (Def)h g = e−1￿ p=1 Cp (De−pf) gg−1p (Dph) gp + f gg−1e Deh ge . Now, by induction hypothesis, the value of the right-hand side at P lies in L · KΛ. On the other hand, the value of the left-hand side at P can be computed as (Def)|2r+2eα− (Def)h g (P ) = ￿ Def g ￿ (P ) ￿ (Def)|2r+2eα Def (P )− h(P ) ￿ = ￿ Def g ￿ (P )λ−2r−n0(λ−2(e+ordP D ef−n0) − 1). Since e > n0 and Def is analytic at P , e+ ordP Def − n0 > 0 and therefore λ−2r−n0(λ−2(e+ordP D ef−n0) − 1) ∈ L ·KΛ ￿ {0}, which yields the result. 190 Automorphic form expansions: arithmetic properties Corollary 5.1. Let f ∈ A0(Γ)L·KΛ and fix P a CM point by Λ ⊂ OK . If g ∈ A0(Γ)L is analytic at P and w ∈ Z≥0 is such that (Dwg)(P ) ￿= 0, then for every e ∈ Z≥0, ((Dwg)−e(Dwef))(P ) ∈ L ·KΛ. Proof. Let g1 ∈ A2w(Γ)L·KΛ such that g1(P ) ￿= 0. Note that it exists because (Dwg)(P ) ￿= 0. Then, the theorem yields Dwg g1 (P ) ∈ (L ·KΛ)∗, and also Dweg ge1 (P ) ∈ L ·KΛ. Therefore, Dwef (Dwg)e (P ) = ￿￿ Dwg g1 ￿−e Dwef ge1 ￿ (P ) ∈ L ·KΛ. Corollary 5.2. Let P be a CM point by an order Λ ⊂ OK . Then, there exists κ = κP ∈ C∗ such that for all f ∈ A2r(Γ)L·KΛ , f = (q￿)r ￿ n anq n, q(z) = κ z − P z − P , an ∈ L ·K Λ. Moreover, κπ−1DK ∈ Q, for πDK = |DK |￿ m=1 Γ ￿ m |DK | ￿￿DK m ￿s , where s =  2, if K = Q(i), 3, if K = Q( √−3), 1/h(K), otherwise. 5.2 Arithmeticity principle 191 Proof. It is enough to prove the result in the case r = 0, because any form of weight 2r can be written as (Dt1)rt2 for some functions t1, t2. We can further assume that f is holomorphic at P . Let q1 = z−Pz−P . Then, every function in A0(Γ)L·KΛ holomorphic at P admits an expansion in C[[qw1 ]], where w is the elliptic order of P . We can choose t ∈ A0(Γ)L·KΛ such that t = a1qw1 + . . . , a1 ￿= 0. Then, define κ = w√a1 and q = κq1. Thus, t = qw + . . . and, if t1 = ￿ n bnq wn, then it follows from Corollary 5.1 that ￿ Dwet1 (Dwt)e ￿ (P ) = bn ∈ L ·KΛ. Therefore, t1 admits an expansion in L ·KΛ[[qw]]. The fact that κπ−1DK ∈ Q follows from [BT08] and [Shi79]. Theorem 5.2. Let P be a CM point by Λ ⊂ OK , let κ ∈ C∗ be as in the previous corollary and let q be the corresponding local parameter. Let also f ∈ A2r(Γ). Then, f ∈ A2r(Γ)L·KΛ if and only if f = (q￿)r ￿ n anq n with an ∈ L ·KΛ for all n. Proof. The rightwards implication follows from the previous corollary. To prove the other, let f ∈ A2r(Γ) be such that f = (q￿)r ￿ n anq n with an ∈ L · KΛ. Moreover, f can be written as f = (Dt1)rt2 for some t1 ∈ A0(Γ)L·KΛ and t2 ∈ A0(Γ). Thus, we can assume that r = 0. In this situation, let {fi}i be a C-base of A0(Γ) with elements lying in A0(Γ)L·KΛ , which exists since A0(Γ) = (A0(Γ)L·KΛ)⊗ C. Therefore, t2 = ￿￿ i aifi, ai ∈ C uniquely. Reordering the functions, we can assume t2 = a1f1 + · · · + anfn uniquely with f1, . . . , fn ∈ {fi}i. Now, since the q-expansions of all these functions have coefficients in L ·KΛ, the constants ai ∈ L ·KΛ. Corollary 5.3. Let f ∈ A0(Γ)L. Then Da(f, z) ∈ A0(Γ)L, where Da(f, z) denotes the automorphic derivative of f introduced in Section 3.3, namely, Da(f, z) = 2f ￿f ￿￿￿ − 3(f ￿￿)2 (f ￿)4 , where f ￿ = dfdz . Proof. Let P be a CM point and Λ ⊂ OK its corresponding order. Let also κP be a local constant corresponding to this point, and qP the corresponding local parameter. Then, f has an expansion lying in L · KΛ[[qP ]] and therefore also 192 Automorphic form expansions: arithmetic properties Da(f, z). Consequently, the previous theorem yields that Da(f, z) ∈ A(Γ)L·KΛ . Therefore, we only need to prove that ￿ P L·KΛ = L. We are going to prove that this intersection, when running over a much more restrictive family of points is already L. Let K = Q(√−p), p > 0, p ≡ 3 (mod 4). Eichler Embedding Theorem ensures that if ￿ −p q ￿ = −1 for every q|DH, then K embeds into H. Moreover, if ￿ −p q ￿ = 1 for every q|N , there exists an optimal embedding of the ring of integers into the Eichler order of level N . Then, the Theorem of Primes in Arithmetic Progression grants us the exis- tence of infinitely many primes satisfiying the above hypotheses. Note that the condition p ≡ 3 (mod 4) excludes no cases, since ￿ ·2￿ has order 8. We choose two such primes, and let K1,K2 be its corresponding quadratic imaginary fields, K(1)1 ,K (1) 2 its respective Hilbert Class Fields, and K˜ (1) 1 , K˜ (1) 2 its Galois closures over Q. Let L˜ be the Galois closure of L|Q. Now, since the ramification sets of K˜(1)1 , K˜ (1) 2 and L˜ are disjoints and these extensions are Galois, they are linearly disjoint. And the result now follows immediately. 5.2.1 Shimura reciprocity In Theorem 1.18 we gave an explicit version of Shimura reciprocity law. This theorem gives an explicit description of the action ofGal(KΛ|K) in the set of CM points by the order Λ. Equivalently, this can be reinterpreted as a description of the action of Gal(KΛ|K) on the values of the functions in A0(Γ)Q at the CM points by the order Λ. Our goal in this section is to prove a similar result not only for the values of functions but for the whole set of coefficients of expansions of quaternionic automorphic forms. The proof is similar to the one we have given to find the field where those coefficients lie, we only need to realize that this reciprocity is compatible with the construction in the proof given before. Lemma 5.5. Let P be a CM point by Λ ⊂ OK and φ : Λ ￿→ O(D,N) an optimal embedding corresponding to P . Let σ ∈ Gal(KΛ|K) and ρ ∈ O(D,N) the element corresponding to σ by Shimura reciprocity law. Then, if α is the one given by Proposition 5.3, O(D,N) ∩ (ραρ−1)−1O(D,N)(ραρ−1) is an Eichler order of level equal to the one of O(D,N) ∩ α−1O(D,N)α and ρ−1P is CM by Λ ⊂ OK with corresponding optimal embedding φρ : Λ ￿→ O(D,N) ∩ (ραρ−1)−1O(D,N)(ραρ−1), 5.2 Arithmeticity principle 193 such that φρ = ρ−1φρ. Proof. Shimura reciprocity law yields that φρ : Λ ￿→ O(D,N) is optimal and its fixed point is ρ−1P . However, by the properties of α, this embedding factors through O(D,N)∩ (ραρ−1)−1O(D,N)(ραρ−1). Moreover, since ραρ−1 satisfies all the properties we required to α, it is an Eichler order of the desired level. Theorem 5.3. Let P be a non-elliptic CM point by Λ ⊂ OK . Consider also f ∈ A2k(Γ)L·K holomorphic at P and g ∈ A2r+2e(Γ)L·K holomorphic and non- vanishing at P . Then, for every σ ∈ Gal(KΛ|K),￿ Def g (P ) ￿σ = Def g (Pσ). Proof. We will proceed by induction on e. The case e = 0 is just standard Shimura reciprocity law, Theorem 1.18. We will begin by showing in detail how to prove the case e = 1 and then we will sketch the general case. Let h = f |2kf ∈ A0(Γ∩α−1Γα). Hence, (Df)h+f(Dh) = (Df)|2k+2α, which evaluated at P yields ￿ Df g ￿ (P )C(λ) = f(Dh) g (P ), with C(λ) ∈ (L · K)∗. Therefore, since the right-hand side is an element of A0(Γ ∩ α−1Γα) evaluated at a CM point P , the case e = 0 applies and we obtain ￿ Df g (P ) ￿σ C(λ) = ￿ Df g (P )C(λ) ￿σ = ￿ fDh g (P ) ￿σ = fDh g (Pσ). Now, define F = f |2kρ, G = g|2kρ and H = F |2kαF , and as above (DF )H G (P ) + F (DH) G (P ) = (DF )|2k+2α G (P ). Moreover, H(P ) = f |2k(ραρ −1) f (ρP ) ∈ (L · K)∗ and indeed is equal to h(P ). Therefore, DF G (P )C(λ) = F (DH) G (P ). 194 Automorphic form expansions: arithmetic properties Putting together the equalities we have found, we obtain the result. In the case e > 1, we keep the notations of the previous case. Then it holds that Def |2r+2eα− (Def)h g = e−1￿ p=1 Cp De−pf gg−1p Dph gp + f gg−1e Deh ge , for gp ∈ A2p(Γ)L·K not vanishing at P (which is possible since P is not elliptic). Now, evaluation at P , using induction hipothesis on the right-hand side and putting this together with the above formula for F,G,H, yields the result. Chapter 6 Computational applications: kroneckerian polynomials and expansions of automorphic functions and forms In this last chapter we study how to compute some of the objects introduced in the previous chapters and we provide several examples. Moreover, additional applications are given as well. In the first section we devote ourselves to the explicit computation of kro- neckerian polynomials and we give some examples for the case D = 6. In the second section, we focus on the computation of the values of a Hauptmodul at a given CM point and we use this to compute all the rational and quadratic CM points of the curves X(D, 1)+, D = 6, 10, 22. In particular, we are able to give another proof that the estimated values for the rational CM points of X(D, 1)+, D = 6, 10, given in [Elk98] are correct (cf. [Err11] for a different ap- proach). In the next section we compute as well the first terms of a q-expansion (those necessary to obtain the rest using the differential equation given by the automorphic derivatives in Theorem 3.1) around all the previously computed rational CM points for X(D, 1)+, D = 6, 10, 22, together with estimates for the corresponding local constants. Finally, in the last section, we focus on the 195 196 Computational applications: kroneckerian polynomials and expansions of automorphic functions and forms study of automorphic forms, we provide generators of the Q-algebras of cusp forms corresponding to the curves considered in Chapters 3 and 4. Moreover, we study how to compute the action of the Hecke operators and give examples of q-expansions for some of the eigenforms. 6.1 Computation of kroneckerian polynomials Let X(D,N) be a Shimura curve such that its quotient by the Atkin-Lehner subgroup W ⊂ WDN has genus 0 (cf. Proposition 4.1). Assume that we know a Hauptmodul t for X(D,N)W , and by known we mean that both the values of the Hauptmodul at the elliptic cycles and the corresponding automorphic derivative are known. In particular, given an elliptic point P of elliptic order e, we can compute q-expansions for the function t = ￿ i aiq ei P with qP = κP z−P z−P for a certain κP ∈ C∗ such that a1 = 1. In this section we are going to see how this information can be used to explic- itly compute kroneckerian polynomials of a given level, under some assumptions on the levels considered, and we will provide some examples. In the next section we are going to find a possible way to overcome this limitation. Fix P ∈ H an elliptic point by Γ = ￿Γ(D,N),W ￿. In particular, P is a CM point by a quadratic order Λ, where Λ is either Z[i], Z[ 1+ √−3 2 ] or is one of the orders appearing in the list of CM points fixed by ωm ∈W , according to Theo- rem 4.1. Denote by φ ∈ E∗(O(D,N),Λ) an optimal embedding corresponding to P . Then, assume that there exists an element λ ∈ Λ￿ of normmk such thatm|λ2 in Λ, for some m such that ωm ∈ W and k such that gcd(k,DN) = 1. If m is not squarefree, assume further that wmφ(λ) is divisible by m in O(D,N). Now, it follows from Proposition 2.1 and Proposition 2.3, that φ(λ) = wmγ0wkγ1, for certain γ0, γ1 ∈ O(D,N)1 and representatives wm, wk of the Atkin-Lehner involutions of levels m, k. In particular, t ◦ φ(λ) = t ◦ (wkγ1) is one of the zeros of the polynomial ΨΓ,k,t(t,X). On the other hand, if we consider a q-expansion of t, t = ￿ i aiq i, since φ(λ) fixes P we have that q ◦ φ(λ) = ￿q, where the expression for ￿ can be found in Proposition 3.6. Therefore, we automatically obtain a q-expansion for the function t ◦ (wkγ1). In particular, we can consider a symmetric polynomial in two variables, of degree p+1 in each variable and indeterminate coefficients, and then use the q-expansions of t and t ◦ (wkγ1) in order to obtain a linear system for the unknown coefficients of this polynomial, when we impose that it vanishes at (t, t ◦ (wkγ1)). 6.1 Computation of kroneckerian polynomials 197 There are formulas to obtain the kroneckerian polynomial of a certain level N from the ones having levels the prime factors of N . For example, the clas- sical formulas for the modular polynomial of level pr (cf. [Cox89],[Web08]) also hold for the kroneckerian polynomials. Namely, if we denote by Ψm(X,Y ) the kroneckerian polynomials for a certain group of quaternionic units and a certain Hauptmodul, then Ψpr (X,Y ) =  ResZ(Ψp(X,Z),Ψpr−1(Y, Z)) Ψpr−2(X,Y )p , if r > 2, ResZ(Ψp(X,Z),Ψp(Y, Z)) (X − Y )p+1 , if r = 2. Similarly, the same is true for the formula of a kroneckerian polynomial of level nm, gcd(n,m) = 1, in terms of the kroneckerian polynomials of levels n,m: Ψnm(X,Y ) = ResZ(Ψn(X,Z),Ψm(Y, Z)). All these expressions involve essentially resultants, which makes them difficult to apply in a setting where the polynomials have high degree and large coeffi- cients. However we will see in the next section how we can use them to compute explicitly the values of a Hauptmodul at certain points. In order to do so, the following results will be useful. Proposition 6.1. Assume that n,m are positive integers with gcd(n,m) = 1 or that n = m = p is a prime number. Then, Ψnm(X,X) = ￿ ResZ(Ψn(X,Z),Ψm(X,Z)), if gcd(n,m) = 1, cResZ(Ψp(X,Z), ∂ ∂X (Ψp(X,Z))), if n = m = p, for a certain c ∈ Z. Proof. The first statement follows immediately from the expression for the kro- neckerian polynomial above. In order to prove the second one, observe that we can write Ψp(X,Z) = Ψp(Y, Z) + (X − Y )Q(X,Y, Z), for a certain polynomial Q(X,Y, Z) such that Q(X,X,Z) = ∂∂X (Ψp(X,Z)). Then, ResZ(Ψp(X,Z),Ψp(Y, Z)) = c(X) αResZ(Ψp(X,Z)−Ψp(Y, Z),Ψp(Y, Z)) 198 Computational applications: kroneckerian polynomials and expansions of automorphic functions and forms where c(X) denotes the leading coefficient of Ψp(X,Z) as a polynomial in Z and α = degZ P (X,Z)− degZ(P (X,Z)− P (Y, Z)). Note that, if c(X) is non-constant, then α = 0. Anyway, c(X)α = c ∈ Z. On the other hand, ResZ(Ψp(X,Z)−Ψp(Y, Z),Ψp(Y, Z)) = ResZ((X − Y )Q(X,Y, Z),Φp(Y, Z)) = ResZ((X − Y ),Ψp(Y, Z))ResZ(Q(X,Y, Z),Ψp(Y, Z)) = (X − Y )p+1ResZ(Q(X,Y, Z),Ψp(Y, Z)). Therefore, Ψp2(X,Y ) = cResZ(Q(X,Y, Z),Ψp(Y, Z)) and the expression on the statement follows after taking X = Y and reordering the factors. Proposition 6.2. Let X(D,M)W be a Shimura curve of genus 0 with a Haupt- modul t and fix a prime p coprime to DM . Let Λ0 be an order inside a quadratic imaginary field K of conductor coprime to p and consider the or- der Λ = Z + pkΛ0. Assume that P ∈ H is a CM point by Λ and denote by φ a corresponding embedding. Assume that there exists λ ∈ Λ￿ ∩ pkΛ0 of norm mNp2k for an integer N coprime to DMp and m￿DM such that ωm ∈ W and, if m is not squarefree, wmφ(λ) ∈ mO(D,M). Then, there exists Q a CM point by Λ0 such that ΨΓ,pk,t(t(P ), t(Q)) = 0. In particular, t(Q) is a root of ΨΓ,N,t(X,X). Proof. Thanks to Propositions 2.1 and 2.3, we can write φ(λ) = γ wpkαwpkγ for a certain α ∈ wmO(D,M)￿N , and γ ∈ O(D,M)1. In particular, α fixes ωpkγP , which is a CM point by Λ0, since it is conjugated to P by an element of norm pk and the corresponding order contains the element λ/pk. Now, by construction of the kroneckerian polynomial, the result follows. Finally, in order to find the value of the Hauptmodul t of the curveX(D,M)W at a CM point by Λ, P , which is fixed by a transformation attached an element λ ∈ Λ￿ of norm mN , with m￿DM but ωm /∈ W , in the conditions of the pre- vious proposition, we could consider the quotient X(D,M)￿W,ωm￿. Then, we could compute the value of a Hauptmodul t￿ at P using Proposition 6.2 and deduce the value of t at P from that of t￿. However, the same proof of Proposi- tion 2.5 can be used to obtain the following result, which will be useful in our computations. 6.1 Computation of kroneckerian polynomials 199 Proposition 6.3. Let W be a subgroup of WDM , Γ = Γ(D,M)￿W ￿ and t a Hauptmodul for the curve X(D,M)WC . Denote S = {s : ωs ∈ W} and let m￿DM . Consider Q(X) = Q0(X)Q1(X) ∈ C(X), with Qi(X) polynomials of degree 1, such that t◦ωm = Q(t). Then, given P a CM point by an order Λ in a quadratic field K, the multiplicity of t(P ) as a root of Q1(X)ψ(N)ΨΓ,N,f (X,Q(X)) is given by r(Λ, N, S) = #{λ ∈ Λ￿ : ωsωmφ(λ) ∈ smO(D,M)N for a certain s ∈ S}/ ∼ = #{λ ∈ Λ￿ : ωsωmφ(λ) ∈ smO(D,M)N for a certain s ∈ S}/e(P ), where λ0 ∼ λ1 if and only if λ0λ1 ∈ NΛ, and e(P ) denotes the elliptic order of P . In particular, the point z0 ∈ C is a root of the polynomial Qψ(N)1 ΨΓ,N,t(X,Q(X)) ∈ C[X] if and only if z0 = t(P ) for a CM point P ∈ H which corresponds to an optimal embedding φ : Λ ￿→ O(D,M) such that there exists a primitive ele- ment λ ∈ Λ of norm smN for a certain s ∈ S with ωsωmφ(λ) ∈ smO(D,M) (cf. Proposition 2.3). Now we are going to present some concrete examples of kroneckerian poly- nomials computed using the method above. We will choose some of the simpler cases, which are D = 6 and N = 5, 7, 11, 13. 6.1.1 Examples in the case D = 6 Let us consider the Hauptmodul t = t6 for Γ = Γ(6, 1) given in Theorem 2.4, further studied in Theorem 3.1. The case p = 5 The point P = i ∈ H is a CM point by Z[i] and therefore it is an elliptic point of order 2 for Γ(6, 1). Denote by φ the corresponding optimal embedding. Moreover, there exists an element λ = 2 + i ∈ Z[i] of norm 5. Thus, for qi(z) = κi z − i z + i , κi ∈ C, 200 Computational applications: kroneckerian polynomials and expansions of automorphic functions and forms we have qi(φ(λ)) = 5 λ2 qi = λ λ qi. Then, we have an expansion t6(z) = ￿ n≥−2 anqi(z) n, where the first few coefficients an ∈ Q are computed explicitly. This gives us t6(φ(λ)(z)) = ￿ n≥−2 anqi(φ(λ)(z)) n = ￿ n≥−2 a￿nqi(z) n, where a￿n = an ￿ 2+i 2−i ￿n . Therefore, we can substitute both series for X, Y in a polynomial ΨΓ,5,t6(X,Y ) = 6￿ j,k=0 ajkX jY k, with indeterminate coefficients ajk, and solve the resulting system. The non-zero coefficients of this polynomial are listed in Table 6.1. It is a trivial verification that the reduction modulo 5 of this polynomial is (X + Y 5)(Y +X5), as stated in Theorem 2.5. a02 = a20 = a46 = a64 119574225 a04 = a40 = a26 = a62 −210039480 a06 = a60 92236816 a11 = a55 66961566 a13 = a31 = a35 = a53 −582922980 a15 = a51 550309200 a22 = a44 −785133000 a24 = a42 1102896150 a33 1712753300 Table 6.1: Coefficients of ΨΓ,5,t6 6.1 Computation of kroneckerian polynomials 201 The case p = 13 This case is similar to the previous one since λ = 3 + 2i ∈ Z[i] has norm 13. Therefore we can take the point P = i ∈ H as above. Then, qi ◦ φ(λ) = 3 + 2i 3− 2i qi. Moreover, from the above expansion for t6 around P = i we obtain t6(φ(λ)(z)) = ￿ n≥−2 anqi(φ(λ)(z)) n = ￿ n≥−2 a￿nqi(z) n, where a￿n = an ￿ 3+2i 3−2i ￿n . Therefore, we can substitute both series for X, Y in a polynomial ΨΓ,13,t6(X,Y ) = 14￿ j,k=0 ajkX jY k, with indeterminate coefficients ajk, and solve the resulting system. The non-zero coefficients of this polynomial are listed in Table 6.2. It is a trivial verification that the reduction modulo 13 of this polynomial is −(X − Y 13)(Y − X13), as stated in Theorem 2.5. a0,2 = a2,0 = a12,14 = a14,12 157321474567113614046314215229221867776 a0,4 = a4,0 = a10,14 = a14,10 −223329048165612085653790111234085115168 a0,6 = a6,0 = a8,14 = a14,8 639558329083494880075337606636408903937 a0,8 = a8,0 = a6,14 = a14,6 −339885081390651885727867107320969704176 a0,10 = a10,0 = a4,14 = a14,4 457846149604433581004932521042517595232 a0,12 = a12,0 = a2,14 = a14,2 102941842560971611003135897276198473984 a0,14 = a14,0 5310433027391928548082272255815434496 a1,1 = a13,13 221553319212858225698359664050620145152 a1,3 = a3,1 = a11,13 = a13,11 −3907822344001165260505767172808382272640 a1,5 = a5,1 = a9,13 = a13,9 5358810648261307600800023159686628064186 a1,7 = a7,1 = a7,13 = a13,7 14188639516967683311243184784903680944504 a1,9 = a9,1 = a5,13 = a13,5 −8509532670217823471919483578243457096096 a1,11 = a11,1 = a3,13 = a13,3 1763597638222966767233053435039160321664 a1,13 = a13,1 −709719245999343588656896320744946706432 a2,2 = a12,12 −5582267572500765247322729910827058588864 a2,4 = a4,2 = a10,12 = a12,10 15566671962647972437246849875069036512271 202 Computational applications: kroneckerian polynomials and expansions of automorphic functions and forms a2,6 = a6,2 = a8,12 = a12,8 17788682738348192950267058609087486547792 a2,8 = a8,2 = a6,12 = a12,6 44512203475303030585138782936649232745468 a2,10 = a10,2 = a4,12 = a12,4 −45481786563481670680883365110716938016016 a2,12 = a12,2 10772996380747037651874095877574912677184 a3,3 = a11,11 22667327663068646397348695143993699978860 a3,5 = a5,3 = a9,11 = a11,9 103292709255177545224740743281321684248456 a3,7 = a7,3 = a7,11 = a11,7 −141385872491557640165418198331988931171624 a3,9 = a9,3 = a5,11 = a11,5 156229489001319601347721493609328430457544 a3,11 = a11,3 −46932210288112374010605471441783240964288 a4,4 = a10,10 123613067875122986869852989122780505527616 a4,6 = a6,4 = a8,10 = a10,8 −234603178607061580963128257263524955945884 a4,8 = a8,4 = a6,10 = a10,6 151963375795261173964103278270125202052688 a4,10 = a10,4 104364071921642313047444412533434295993990 a5,5 = a9,9 −460109713337629190182764691769863407901680 a5,7 = a7,5 = a7,9 = a9,7 545160957003294505782278920619623236703800 a5,9 = a9,5 −316049176219180561461317385354087527471140 a6,6 = a8,8 326166687776446607181743146779777359568000 a6,8 = a8,6 −460420624875331284131007926726632542448230 a7,7 −1086940199760004041936334880320578594960120 Table 6.2: Coefficients of ΨΓ,13,t6 The case p = 7 Now, the point P = 1+i 1+ √ 3 ∈ H is a CM point by the ring of integers of Q(√−3) and thus an elliptic point of order 3 for Γ(6, 1). Moreover, there exists an element λ = 2 + √−3 ∈ Z[ 1+ √−3 2 ] of norm 7. Then, if we denote by φ the optimal embedding corresponding to P , we have that qP ◦ φ(λ) = 2 + √−3 2−√−3qP and therefore, given the expansion t6(z) = ￿ n≥0 anqP (z) n, 6.1 Computation of kroneckerian polynomials 203 where the first few coefficients an ∈ Q are computed explicitly, we obtain an expansion for t6(φ(λ)(z)) = ￿ n≥0 anqP (φ(λ)(z)) n = ￿ n≥0 a￿nqP (z) n, where a￿n = an ￿ 2+ √−3 2−√−3 ￿n . Therefore, we can substitute both series for X, Y in a polynomial ΨΓ,7,t6(X,Y ) = 8￿ j,k=0 ajkX jY k, with indeterminate coefficients ajk, and solve the resulting system. The non-zero coefficients of this polynomial are listed in Table 6.3. It is a trivial verification that the reduction modulo 7 of this polynomial is (1−XY 7)(1−Y X7), as stated in Theorem 2.5. a00 = a88 366028679279376 a02 = a20 = a68 = a86 −175148154031032 a04 = a40 = a48 = a84 83697102786465 a06 = a60 = a28 = a82 −15011938285344 a08 = a80 2688917803264 a11 = a77 −2484877824079056 a13 = a31 = a57 = a75 2326773665288268 a15 = a51 = a37 = a73 632022013171836 a17 = a71 −193115439357632 a22 = a66 669210475564134 a24 = a42 = a46 = a64 −3793573444895640 a26 = a62 3577214930233798 a33 = a55 9188300872159416 a35 = a53 −12672039590642936 a44 6858140981941060 Table 6.3: Coefficients of ΨΓ,7,t6 The case p = 11 This case is more complicated since neither Z[i] nor Z[ 1+ √−3 2 ] contain elements of norm 11. However, if we move to the quotient X(6, 1)+, the CM points by 204 Computational applications: kroneckerian polynomials and expansions of automorphic functions and forms Z[ √−6] have become elliptic and therefore we can compute an expansion of t+6 around the CM point by Z[ √−6] given by P = ( √ 6−√2)i 2 , where t + 6 (P ) = 0. From this expansion we obtain an expansion for t6 around the point P , where t6(P ) = i, using that t+6 = ￿ t26 + 1 t26 − 1 ￿2 . The ring Z[ √−6] does not contain any element of norm 11, but it does con- tain an element of norm 22, namely λ = 4 + √−6, such that 2|λ2 in Z[√−6]. Therefore we could use the procedure above to compute the kroneckerian poly- nomial of level 11 attached to t+6 or even for the Hauptmodul corresponding to X(6, 1)￿ω2￿, u6,3. However, since we know explicitly the action of the Atkin- Lehner involutions on t6, in our case t6 ◦w2 = −t6, we can directly compute the kroneckerian polynomial of level 11 corresponding to t6, using Proposition 6.3. Note that t6 ◦ φ(λ) = t6(ω2γ0w11γ1) = −t6(w11γ1) and therefore we have that t6(w11γ1) = −t6 ◦ φ(λ). Then, for qP (z) = κP z − P z − P , κP ∈ C ∗ we have qP (φ(λ)) = λ λ qP . Moreover, we have an expansion t6(z) = ￿ n≥0 anqP (z) n, where the first few coefficients an ∈ Q are computed explicitly. This gives us a relation t6(w11γ1(z)) = ￿ n≥0 a￿nqP (z) n, where a￿n = −an ￿ 4+ √−6 4−√−6 ￿n . Therefore, we can substitute these series￿ n≥0 anqP (z) n, ￿ n≥0 a￿nqP (z) n, for X, Y in a polynomial ΨΓ,11,t6(X,Y ) = 12￿ j,k=0 ajkX jY k, 6.1 Computation of kroneckerian polynomials 205 with indeterminate coefficients ajk, and solve the resulting system. The non-zero coefficients of this polynomial are listed in Table 6.4. It is a trivial verification that the reduction modulo 11 of this polynomial is (1 +XY 11)(1 + Y X11), as stated in Theorem 2.5. a0,0 = a12,12 161479346678868884449383016704 a0,2 = a2,0 = a10,12 = a12,10 −700259488320028961583165392640 a0,4 = a4,0 = a8,12 = a12,8 919912311217388018171537308512 a0,6 = a6,0 = a6,12 = a12,6 76545318714518090586092693904 a0,8 = a8,0 = a4,12 = a12,4 −881662021257852546155119386711 a0,10 = a10,0 = a2,12 = a12,2 211560066253894917848033517696 a0,12 = a12,0 279732396586128790436714745856 a1,1 = a11,11 −5152213998401374154602359608832 a1,3 = a3,1 = a9,11 = a11,9 18511567572675167012497768926336 a1,5 = a5,1 = a7,11 = a11,7 −25899527501613233338738322103264 a1,7 = a7,1 = a5,11 = a11,5 28376635514179567967013595298904 a1,9 = a9,1 = a3,11 = a11,3 −23743359785030634825102696625956 a1,11 = a11,1 7438491727516340412718850925312 a2,2 = a10,10 59328230855330299186274825945664 a2,4 = a4,2 = a8,10 = a10,8 −164185592320146228724455084990864 a2,6 = a6,2 = a6,10 = a10,6 201983865932424685349472770205756 a2,8 = a8,2 = a4,10 = a10,4 −125853332231546454366142736028912 a2,10 = a10,2 31648366312546597884262004848374 a3,3 = a9,9 −316120114238429618184282492087360 a3,5 = a5,3 = a7,9 = a9,7 646552456939854174209188752276840 a3,7 = a7,3 = a5,9 = a9,5 −544407496253705110435610320239120 a3,9 = a9,3 211652433379326599389385977311760 a4,4 = a8,8 905214213307076899036139519181750 a4,6 = a6,4 = a6,8 = a8,6 −1330514179243565505915761972442000 a4,8 = a8,4 733895948770622934801403475408040 a5,5 = a7,7 −1772204569903587914287480135652760 a5,7 = a7,5 1637057782264734851739512604044400 a6,6 2296209946830565387198646731768260 Table 6.4: Coefficients of ΨΓ,11,t6 206 Computational applications: kroneckerian polynomials and expansions of automorphic functions and forms 6.2 Computation of the values of a Hauptmodul and its expansions at a given CM point First of all, given a Hauptmodul t for a genus 0 curve X(D,N)W corresponding to the group Γ = ￿Γ(D,N),W ￿, we are going to describe a method to compute the value of t around an arbitrary CM point using only the expansions around elliptic points and the corresponding kroneckerian polynomials we can compute from them according to the previous section. Then we are going to see how to extend this method to compute as well expansions around these points. Note that in this way we can go back to the previous section and use these new ex- pansions to compute new kroneckerian polynomials which we could not compute directly using only the expansions at elliptic points. Assume we are given a non-elliptic CM point P by an order ΛK of conductor f in a quadratic imaginary field K and denote by φK a corresponding optimal embedding. Then we need to compute a0, a2 ∈ Q such that t = a0+q+a2q+ . . . for a certain local parameter q = κ z−P z−P . If these two values are known, then we can use the expression of the automorphic derivative to compute as many coefficients of the q-expansions around P as desired. In order to compute this couple of algebraic numbers we make the following choices: • An elliptic point Q by an order ΛF in a quadratic imaginary field F . Denote by φF an optimal embedding corresponding to Q. • An odd prime p such that: – p decomposes in both F and K; – gcd(p, f) = 1. Then, there exists λF ∈ Λ￿F of norm s0pr0 for some positive integers s0, r0 such that ωs0 ∈W , λ2F ∈ s0ΛF and, if s0 is not squarefree, we require in addition that ωs0φF (λF ) ∈ s0O(D,N) (cf. Proposition 2.3). Similarly, there exists λK ∈ Λ￿K of norm s1pr1 for some positive integers s1, r1 such that ωs1 ∈ W , λ2K ∈ s1ΛK and, if s1 is not squarefree, we have as well that ωs1φK(λK) ∈ s1O(D,N). Then, we define r = lcm(r0, r1). Since we will be looking for an r which is as small as possible, it will be convenient in most cases to take s0 = r0 = 1 (for 6.2 Computation of the values of a Hauptmodul and its expansions at a given CM point 207 the existence of such primes in this case, cf. [Cox89]). Note that some choices of F and p can lead to more computationally complex situations than others. Let ΨΓ,pr,t(X,Y ) ∈ C[X,Y ] be the kroneckerian polynomial of level pr at- tached to a Hauptmodul t. By construction, this polynomial can be computed using expansions around the elliptic point Q. Since P is not elliptic, we can fix a local constant κ = κP ∈ C and a local parameter q = qP (z) = κ z−Pz−P such that the expansion of t around P with respect to this parameter is t(q) = a0 + q + a2q 2 + . . . Now, since α = φF (λF ) fixes P , q◦αq = ￿ ∈ C∗ and therefore (t ◦ α)(q) = a0 + ￿q + ￿2a2q2 + . . . By definition, ΨΓ,pr,t(t, t ◦ α) = 0 and thus a0 is a zero of ΨΓ,pr,t(X,X). More details on the zeros of ΨΓ,pr,t(X,X) and their multiplicity are given in Propo- sition 2.5. This fact, together with an approximation of the value at P , if necessary, allows us to compute the exact value of a0. In order to compute the value of a2, we start by observing that due to the choices above, Theorem 2.3 gives us that mult(P,ωprγP )ΨΓ,pr,t = 2, where γ is such that φK(λ r/r0 K ) = ω r/r0 s0 γ ￿wprγ. In particular, since φK(λ r/r0 K ) fixes P , P defines the same point as ωprγP in ￿Γ(D,N),W ￿\H. Therefore, we obtain that mult(P,P )ΨΓ,pr,t = 2 and, by Theorem 2.2, that ordf(P )Ψ(f(P ), X) = 2. In particular, all first order derivatives of ΨΓ,pr,t(X,Y ) vanish at (a0, a0) and there exists a second order derivative of ΨΓ,pr,t(X,Y ) not vanishing at (a0, a0). Using that ΨΓ,pr,t(t(q), t◦α(q)) = 0 and that ΨΓ,pr,t is symmetric we obtain, differentiating twice with respect to q and evaluating at q = 0, (1 + ￿2)(∂XXΨΓ,pr,t)(a0, a0) + 2￿(∂XYΨΓ,pr,t)(a0, a0) = 0. Similarly, differentiating three times and evaluating at q = 0, we obtain an expression containing a2, powers of ￿ and derivatives up to order three of ΨΓ,pr,t evaluated at (a0, a0). The coefficient of a2 in this expression is (1 + ￿3)(∂XXΨΓ,pr,t)(a0, a0) + (￿+ ￿ 2)(∂XYΨΓ,pr,t)(a0, a0). If it is not zero, then we will be able to compute a2 from ￿, a0 and the values of the kroneckerian polynomial. To prove that this expression is really non-zero we need to check that ￿ ￿= 0, 1,−1. By Proposition 3.6, ￿ = λK λK , and therefore 208 Computational applications: kroneckerian polynomials and expansions of automorphic functions and forms it is obvious that ￿ ￿= 0, 1; if ￿ were −1, then λK = √−s1pr1 and since p does not ramify in K, r1 ∈ 2Z and moreover it is primitive in ΛK , therefore ΛK = Z[pr ￿ 1 √−s1] and it has conductor coprime to p from what we obtain a contradiction, namely 0 < r1 = 2r￿1 = 0. Hence, we get the value of a2 as desired. Remark. The above choice of p is convenient to have a general approach to the problem, but it is not always the most suitable approach to our particular case. To compute the value of t at a certain CM point P by ΛK as above, it is enough to know ΨΓ,N,t(X,X) for a certain N such that there exists λK ∈ Λ￿K of norm s1N with ωs1 ∈ W such that λ2K ∈ s1ΛK and, if s1 is not squarefree, ωs1φ(λK) ∈ s1O(D,N). Then, Proposition 2.5 yields that t(P ) is a root of this polynomial. Similarly, to compute the value of a2 it suffices to have ΨΓ,pr,t(X,Y ) for a certain pr with gcd(p, f) = 1 such that there exists λK ∈ Λ￿K of norm s1pr with ωs1 ∈W , λ2K ∈ s1ΛK and, if s1 is not squarefree, ωs1φ(λK) ∈ s1O(D,N). Then, Theorem 2.3, ensures that mult(P,P )ΨΓ,pr,t ≤ 2 and therefore, if the multiplicity is 1 the value of a2 is obtained immediately and if it is 2 we can proceed as described in the method above. 6.2.1 All rational and quadratic CM points on the curves X(D, 1)+ for D = 6, 10, 22 As an application of the previous algorithm we are going to compute the set of rational and (non-rational) quadratic CM points on the curves X(D, 1)+, D = 6, 10, 22. We denote the corresponding group by Γ+. In particular we obtain that the tables for D = 6, 10 of rational CM points given by Elkies in [Elk98], where some of the values were only numerically correct, are indeed correct. There is only one missing value in [Elk98] in the case D = 10. A completely different approach to check that these values in the cases D = 6, 10 are correct can be found in [Err11]. The case D = 22 was not studied in Elkies’ paper, and the other two cases studied there, D = 14, 15, can also be checked using our approach. To do so, in all these cases, we compute the kroneckerian polynomials ΨΓ+,p,t+D (X,Y ) for all primes 3 ≤ p ≤ 41 coprime to D, which can be obtained using the method we have seen. From these, the polynomials ΨΓ(D,1),p,tD (X,Y ) can also be deduced. We do not reproduce all these polynomials, because they are too large to be printed here. 6.2 Computation of the values of a Hauptmodul and its expansions at a given CM point 209 Then, we consider the list of orders which may give rational or quadratic CM points on X(D, 1)+. In particular, all these CM points must be attached to fields having class number 1, 2, 4 or 8. The list of quadratic imaginary fields with class numbers 4 are given in [Arn92] and those of class number 8 can be computed thanks to the bounds given in [Wat04]. The list of such fields is given in the following table: h |D(K)| for imaginary quadratic K with hK = h 1 3, 4, 7, 8, 11, 19, 43, 67, 163 2 15, 20, 24, 35, 40, 51, 52, 88, 91, 115, 123, 148, 187, 232, 235, 267, 403, 427 4 39, 55, 56, 68, 84, 120, 132, 136, 155, 168, 184, 195, 203, 219, 228, 259, 280, 291, 292, 312, 323, 328, 340, 355, 372, 388, 408, 435, 483, 520, 532, 555, 568, 595, 627, 667, 708, 715, 723, 760, 763, 772, 795, 955, 1003, 1012, 1027, 1227, 1243, 1387, 1411, 1435, 1507, 1555 8 95, 111, 164, 183, 248, 260, 264, 276, 295, 299, 308, 371, 376, 395, 420, 452, 456, 548, 552, 564, 579, 580, 583, 616, 632, 651, 660, 712, 820, 840, 852, 868, 904, 915, 939, 952, 979, 987, 995, 1032, 1043, 1060, 1092, 1128, 1131, 1155, 1195, 1204, 1240, 1252, 1288, 1299, 1320, 1339, 1348, 1380, 1428, 1443, 1528, 1540, 1635, 1651, 1659, 1672, 1731, 1752, 1768, 1771, 1780, 1795, 1803, 1828, 1848, 1864, 1912, 1939, 1947, 1992, 1995, 2020, 2035, 2059, 2067, 2139, 2163, 2212, 2248, 2307, 2308, 2323, 2392, 2395, 2419, 2451, 2587, 2611, 2632, 2667, 2715, 2755, 2788, 2827, 2947, 2968, 2995, 3003, 3172, 3243, 3315, 3355, 3403, 3448, 3507, 3595, 3787, 3883, 3963, 4123, 4195, 4267, 4323, 4387, 4747, 4843, 4867, 5083, 5467, 5587, 5707, 5947, 6307 For any of these fields, the set of orders we have to consider can be computed using Theorem 1.12. Then, among this orders we can select those giving rational or quadratic CM points thanks to the following result, cf. [Jor81, GR06]. In [GR06], a version of 210 Computational applications: kroneckerian polynomials and expansions of automorphic functions and forms this result with squarefree level is stated. Since we are interested in the level 1 case, we will restrict to this case. Proposition 6.4. Let X(D, 1) be the Shimura curve of discriminant D. For a quadratic order Λ of conductor f in a quadratic imaginary field K, consider the subgroup of Atkin-Lehner involutions given by W (Λ) = ￿ωp : p ￿ f, ￿ K p ￿ = −1￿. Then, W (Λ) × Gal(KΛ|K) acts freely and transitively in the set of CM points by Λ. Note, in particular, that the CM points by Λ are rational in X(D, 1)+ if and only if there exists only one representative of such points on ￿Γ(D, 1),WD￿\H. The case D = 6 First of all, we have to consider among those fields above, the ones which are embeddable into H6. Then we check which of those yield rational and quadratic CM points. We give them in the following table: f 1 5 7 11 13 |D(K)| s.t. the order of conductor f gives rational CM points 3, 4, 19, 24, 40, 43, 51, 52, 67, 84, 88, 120, 123, 132, 148, 163, 168, 228, 232, 267, 312, 372, 408, 708 3, 4 3 |D(K)| s.t. the order of conductor f gives quadratic CM points 91, 115, 136, 184, 187, 195, 219, 235, 264, 276, 280, 291, 292, 328, 340, 388, 403, 420, 427, 435, 456, 483, 520, 532, 552, 555, 564, 568, 627, 660, 723, 760, 772, 795, 840, 852, 1012, 1032, 1092, 1128, 1227, 1320, 1380, 1428, 1752, 1848, 1992 24 4 3 3 Now that we have the list of CM points for which the values of t+6 need to be computed, we have to use Proposition 2.5 to identify the value of t+6 at each of them as a zero of ΨΓ+,N,t+6 (X,X), for some N such that we can explicitly compute ΨΓ+,N,t+6 (X,X). We can use Proposition 6.1 to deal with the composite levels. In this case, Proposition 6.2 will not be needed. 6.2 Computation of the values of a Hauptmodul and its expansions at a given CM point 211 In Table 6.5, for each order Λ with discriminant D(Λ) such that the points in CM(Λ) are rational in X(6, 1)+, according to the table above, we provide a list of some N such that ΨΓ+,N,t+6 (X,X) has t(P ) as a root, for all points P ∈ CM(Λ). In this list, we do not provide values of N for the rings of integers of Q( √−d), for d = 1, 3, 6, since the values at these points are already known by construction of the Hauptmodul t+6 . D(Λ) N −40 5, 7, 11, 13 −84 5, 7, 11, 17, 19, 23, 27 −19 5, 7, 11, 17, 19, 23 −120 5, 37 −51 5(11) −52 7, 11, 13 −132 7, 11 −168 7, 23 −3 · 52 7(13) −43 11, 13, 17 −88 11, 13 −123 11, 17 D(Λ) N −228 11(19) −312 13, 19, 29 −3 · 72 13, 19 −4 · 52 13(29) −67 17, 19 −372 17, 29 −408 17(37) −148 19(23) −267 23(41) −232 29(31) −708 31 −163 41 Table 6.5: Some N for which ΨΓ+,N,t+6 (X,X) vanishes at t + 6 (CM(Λ)) Moreover, the order in which Λ appears in the list can be used to determine the values of t+6 at all the corresponding CM points. The factorization of all these polynomials can be computed in a reasonable time and therefore the cor- responding values are obtained. For some orders where only one N is needed to determine the value, another one is given into parentheses whenever possible. Since the Hauptmodul t+6 we have considered coincides with the one used in [Elk98], the values we have computed are the same which appear there. They can be found in Table 6.6. 212 Computational applications: kroneckerian polynomials and expansions of automorphic functions and forms D(Λ) t+6 (P ), P ∈ CM(Λ) −3 ∞ −4 1 −24 0 −40 2312125 −84 − 16927 −19 32111024 −120 57763375 −51 − 13771024 −52 687715625 −132 1368915625 −168 − 70178415625 −3 · 52 152881138240 −43 2125098716000000 −88 1554588820796875 D(Λ) t+6 (P ), P ∈ CM(Λ) −123 − 29690072116000000 −228 6686332911390625 −312 2700874238427680640625 −3 · 72 − 10731520813024000000 −4 · 52 4218505211771561 −67 779037006671024000000 −372 − 455413074649747377296875 −408 − 3240860943673655962140625 −148 69630712957377149515625 −267 − 57666817144887211814078464000000 −232 6643227848345223256413239012828125 −708 71475755554842930369224337327397603890625 −163 699690239451360705067684178814003344000000 Table 6.6: Rational values of t+6 at the CM points by the order of discriminant D(Λ) 6.2 Computation of the values of a Hauptmodul and its expansions at a given CM point 213 Similarly, we can reproduce the procedure above to compute the value of t+6 at the quadratic CM points. First of all, we begin by finding values of N such that ΨΓ+,N,t+6 (X,X) vanishes at t + 6 (CM(Λ)); a suitable selection of such values is given in Table 6.7. D(Λ) N −264 11(17) −276 13(23) −136 17, 19 −195 17(23) −219 19, 37 −420 19, 31 −456 19(41) −91 23, 5 · 7 −552 23, 29 −184 23(31) −4 · 72 29, 37 −115 29, 31 −660 29(41) −3 · 112 31, 37 −24 · 52 31, 7 · 11 −291 31(5 · 17) −280 37, 5 · 7 −292 37, 41 −435 37, 5 · 11 −564 37, 5 · 19 −852 37(72) −483 41, 7 · 11 −840 41, 5 · 7 −328 41(7 · 13) −340 5 · 11, 5 · 17 −1320 5 · 11 D(Λ) N −235 5 · 13, 5 · 23, 132 −555 5 · 13, 132 −520 5 · 13(5 · 23) −795 5 · 17 −1380 5 · 19, 5 · 23 −760 5 · 19(5 · 29) −187 7 · 11, 72 −1848 7 · 11 −532 7 · 13, 7 · 19 −1092 7 · 13(7 · 23) −1428 7 · 17, 7 · 19 −427 7 · 17 −3 · 132 7 · 19, 132 −388 7 · 19(72) −1032 7 · 23(72) −627 11 · 13, 13 · 17 −1128 11 · 13(132) ∗ −568 11 · 13(132) ∗ −403 11 · 13(112) −723 11 · 17, 112 −1012 11 · 17 −772 11 · 19(112) −1227 11 · 29(112) −1752 13 · 17(132) −1992 132 Table 6.7: Some N for which ΨΓ+,N,t+6 (X,X) vanishes at t + 6 (CM(Λ)) Moreover, studying the factors of the corresponding polynomials in the order given in the table also allows us to find out which factor corresponds to which 214 Computational applications: kroneckerian polynomials and expansions of automorphic functions and forms order, except for a couple of orders marked in the list. Note as well that for all the orders for which only one polynomial is necessary to find the corresponding quadratic factor, another one is given when available to help in the computation. To determine which quadratic factor corresponds to each of the polynomi- als marked with ∗, we can study the factor of ΨΓ+,11·13,t+6 (X,X) coming from ΨΓ,11·13,t6(X,X), since it vanishes at t + 6 (P ) for P a CM point by the order of discriminant −568 but not for those attached to the order of discriminant −1128. Otherwise, we can consider the extension defined by the possible poly- nomials over the Q(Λ). Since it must be contained in the Hilbert Class Field it has to be unramified and it only happens for one of the choices. Finally, studying the factors of these polynomials we obtain the values of t+6 (P ), for all those quadratic CM points in X(6, 1)+. We give the irreducible polynomials of t+6 (P ) for all these P in Table 6.8. D(Λ) Irr(t+6 (P ), X) for P ∈ CM(Λ) −264 42719296− 53148528X + 26198073X2 −276 −206586207 + 328736070X + 1771561X2 −136 52236012608 + 9087757328X + 8703679193X2 −195 884218628241− 1812440448000X + 1048576000000X2 −219 −128030514913249 + 125096383466496X + 1320903770112X2 −420 3732357388761− 1357411709250X + 377149515625X2 −456 31050765799488− 43342656704496X + 42761175875209X2 −91 86743200841609− 37494741699584X + 118887446216704X2 −552 −5477054994808384 + 3075898907250000X + 2165464599609375X2 −184 384963067803392− 412829614079584X + 293681802023X2 −4 · 72 642574792185481− 402872644381538X + 262254607552729X2 −115 2241128724615668409− 261027932841728000X + 29025255424000000X2 −660 4314790752945768481− 927163113067208250X + 1229255326344515625X2 −3 · 112 22527082795349027281−61407732535776000000X+55427328000000000000X2 −24 · 52 200192797421660160− 25808161838902560X + 3573226485213841X2 −291 −65127370229218609− 383293130981529600X + 36563462560677888X2 −280 14831567427889598464+15754661879632468000X+16465090583970015625X2 −292 19486892954181586609− 198140231857576531250X + 36141574462890625X2 −435 569481201891590025002641 −586157134962017354112000X +18451045077221376000000X2 −564 −389128161774795183 + 391849027410973158X + 262254607552729X2 −852 −14251861243863222453279− 45770840670130176656250X +64027003797052978515625X2 6.2 Computation of the values of a Hauptmodul and its expansions at a given CM point 215 −483 574403839700283504692001 + 5862200747621456160000000X +6179217664000000000000X2 −840 372225887737368128067136− 724882883109089333382000X +355078782317139783890625X2 −328 11389413669422886016064− 972791427994051750000X +406148777599853515625X2 −340 59051908771578859412809− 93199961669184360994250X +44148012183246943140625X2 −1320 59399623661988395071714457480256−127126399712714190946811482218000X +552623574110417481699862828890625X2 −235 98428582062197494896026775289− 49859549361495492862485248000X +17264898824176943104000000X2 −555 5911888183811885288521821681− 3386073613019855693666688000X +3746799534959588540416000000X2 −520 27984106050222502857738439203904− 18961382550718101625425188102000X +5546146627308279494096122515625X2 −795 6706046582828456498517769253814186961 −1600029201681539436338910607033728000X +9353153695832353600224362496000000X2 −1380 22620611769657280039067017482827761 −8917712716009873489898962035989250X +10795739863542751133778915140625X2 −760 452886502050801398701510350192484964948224 −445437509849522458916545214029502351588000X +4725124962498114462443158539086265625X2 −187 1281803527663144053051289 + 146223250516439200000000X +37897187584000000000000X2 −1848 80941128040036959539319606963373261148416 +7566730511853299767505362064493337500000X +1435259088308157779576399434191650390625X2 −532 37392890420940106684440901945609 + 286100070244316021555308093750X +3218720433250754250244140625X2 −1092 881180666024379209413236481− 2072108999454045698560781250X +1497706903482069465087890625X2 −1428 5080723985623053566256573185105841 +2595138632671038174308218113843750X +111004163771598482150634765625X2 −427 236345145177884150466328274020723680186889 −249208777711953197450743796334028592000000X +95235372311230160330277535744000000000000X2 −3 · 132 103672964458079375562609121− 21810524179637804832000000X +4298005400832000000000000X2 216 Computational applications: kroneckerian polynomials and expansions of automorphic functions and forms −388 9912915372060849777348889− 14660860130078731447531250X +4662118527211564697265625X2 −1032 2083334188693550710835870784− 56573541798919546839750000X +2054467631662646728515625X2 −627 104817080583432592888674903921− 159879480719277943213920000000X +67137179533498624000000000000X2 −1128 −4349405099360019567740874304− 5020059266433564105066750000X +10991302018397804443359375X2 −568 59587335617850194720528564631296 −117094519737299049547564100500000X +22735042692560934566677490234375X2 −403 4402921782424728674818063256213417644489 −175120941766664486593304207549360000000X +53689604058838582206398464000000000000X2 −723 −9038300822385717397069614840001 −130975190212253125622047632000000X +139838497004915048448000000000000X2 −1012 2570187045545914742463370102372216333917137449 −89850877541163481662946793921060591396341906250X +87483469110298477677527973845535739898681640625X2 −772 4708627161358049627738157747286403827249 −5224321697045595217012469095718953531250X +498411084409233573904241044746337890625X2 −1227 −44859378464919768563391446276055834785514128161 +21434298178618274731026952272896386716192000000X +85525366214177410333043640607488000000000000X2 −1752 −80858407073956482814358210802559770624 +58949096799548669418385000633426500000X +21991775872076776648979834308837890625X2 −1992 2570187045545914742463370102372216333917137449 −89850877541163481662946793921060591396341906250X +87483469110298477677527973845535739898681640625X2 Table 6.8: Irreducible polynomials of the values of t+6 at the CM points by the order of discriminant D(Λ) 6.2 Computation of the values of a Hauptmodul and its expansions at a given CM point 217 The case D = 10 We follow the same approach which we used in the case D = 6. We begin by finding the orders giving rational and quadratic CM points on the curve X(10, 1)+: f 1 3 7 |D(K)| s.t. the order of conductor f gives rational CM points 3, 8, 20, 35, 40, 43, 52, 67, 88, 115, 120, 148, 163, 232, 235, 280, 340, 520, 760 3, 8, 20 |D(K)| s.t. the order of conductor f gives quadratic CM points 91, 115, 136, 184, 187, 195, 219, 235, 264, 276, 280, 291, 292, 328, 340, 388, 403, 420, 427, 435, 456, 483, 520, 532, 552, 555, 564, 568, 627, 660, 723, 760, 772, 795, 840, 852, 1012, 1032, 1092, 1128, 1227, 1320, 1380, 1428, 1752, 1848, 1992 35, 40 3 Now that we have the list of CM points we have to work with, we find suitable values of N such that the value of t+10 at each of them is a zero of ΨΓ+,N,t+10(X,X), where this polynomial can be explicitly computed. In Ta- ble 6.9, for each order Λ with discriminant D(Λ) such that the points in CM(Λ) are rational in X(10, 1)+, according to the table above, we provide a list of some N such that ΨΓ+,N,t+10(X,X) has t + 10(P ) as a root, for all CM points by Λ, P . We provide as well the corresponding values of the Hauptmodul t+10. Note that in this case our Hauptmodul does not match the one used in [Elk98]. If we denote by f the one there, we have the following relation: f = 2t+10 − 27 t+10 − 1 . Using this we recover all the values given in [Elk98], together with the missing value for t+10(CM(Z[ 1+ √−135 2 ])). Next, we move to the case of CM points were t+10 takes quadratic values and provide the results in Table 6.10. 218 Computational applications: kroneckerian polynomials and expansions of automorphic functions and forms D(Λ) N t+10(CM(Λ)) −3 (3) 272 −8 (3) 1 −20 (3) ∞ −40 (7) 0 −35 3, 7 − 52 −120 3(31) 545 −52 7, 11 729104 −280 7, 17, 19 − 6561845 −3 · 32 7, 13 867242 −115 7, 17 − 3645242 −20 · 32 7(23) 289120 D(Λ) N t+10(CM(Λ)) −43 11, 13, 17 31347722 −88 11, 13 7293179 −8 · 32 11, 19 3844169 −340 11(17) − 1822522984 −520 13, 31 12131289874225 −235 13(23) − 306544587362 −67 17(19) 439587232562 −148 19, 23 5517801342176 −760 19(29) 294523290531898939 −232 29(31) 106571781171374281 −163 41 18300320498715685392962 Table 6.9: Some N for which ΨΓ+,N,t+10(X,X) vanishes at t + 10(CM(Λ)) and value of t+10(CM(Λ)) 6.2 Computation of the values of a Hauptmodul and its expansions at a given CM point 219 D(Λ) N Irr(t+10(P ), X) for P ∈ CM(Λ) −195 11(3 · 7) 700569− 594540X + 36100X2 −260 13, 29 36125− 5120X + 208X2 −68 13, 17 14297 + 1056X + 272X2 −420 13(3 · 7) 210681− 259200X + 115600X2 −132 17, 3 · 7 2712609− 399168X + 327184X2 −580 17, 37 15411789− 1982880X + 338000X2 −35 · 32 17(7 · 13) 89397025− 17972620X + 1387684X2 −40 · 32 19, 23 5400− 1845X + 289X2 −660 19, 3 · 11 5061611025− 344413080X + 8952064X2 −355 19, 29 −37732311− 227783340X + 11628100X2 −155 19(3 · 13) −26071− 60460X + 12100X2 −168 23, 29 72795024 + 3601827X + 165649X2 −435 23, 3 · 11 32871503025− 68044642380X + 39633642724X2 −820 23(41) −2832049089− 370536120X + 43808000X2 −228 29, 41 2503100961 + 2500698528X + 1336341136X2 −555 29, 3 · 13 201138115488489− 33082706982060X + 971190540100X2 −1060 29(7 · 19) 625041046125− 341752050720X + 56700102992X2 −123 31, 37 64126952289− 44678870028X + 397683364X2 −840 31, 3 · 7 145138140900− 6918248205X + 1527480889X2 −1240 31, 11 · 17 −884370968100 + 942728817780X + 20146684519X2 −595 31(41) 1566735366249 + 1473678646740X + 15205356100X2 −292 37, 41 395749763793− 21388556736X + 948183568X2 −1380 37, 3 · 23 187881086227445361− 21968541318636000X + 803465102611600X2 −715 37, 11 · 13 15069341705625− 11139015359340X + 8161866182404X2 −3 · 72 37, 3 · 13 156652098849− 186645543948X + 203778210724X2 −795 41, 3 · 17 706183354362148569− 683747285443080300X + 19465258372202500X2 −312 41, 3 · 13 4616391719241− 1024325855982X + 13185699241X2 −1540 41, 7 · 11 17208167475809025− 2412703826676000X + 85157386661776X2 −328 41(7 · 13) 177545935044− 260353891413X + 19261831369X2 −1320 3 · 11(3 · 37) 4253917632420096 + 222068375568360X + 152919316263025X2 −372 3 · 17, 3 · 29 612210160180161− 422027880219072X + 26809032929536X2 −408 3 · 17, 3 · 23 692582067838116 + 294235182711468X + 42878648216041X2 −267 3 · 23(3 · 29) 8209389990443736249− 1639217860896435948X + 78471426340840324X2 −708 3 · 31 300082949530934595113169 + 3924120402979729948512X +31961284468815788944X2 −187 7 · 11(7 · 17) 13005854087769− 2684967194988X + 553443747844X2 −532 7 · 13(7 · 19) 69021381726708849− 18231570739375848X + 1604357522842624X2 −427 7 · 17(172) 1864064969173422228249− 60288183798432272748X + 511922609014091685124X2 −388 7 · 19, 7 · 23 92700984533337− 17805086855424X + 1109805837712X2 −1435 7 · 19 3459273912303717380625 + 88188079891115816674740X +370557220051807928164X2 −568 11 · 13, 132 8285001402009564− 12863267765246628X + 3482283956596439X2 −403 11 · 13, (112) 2165643130620871566729− 219064717977585969708X + 612006435839793604X2 −1012 11 · 17(11 · 23) 2085239189998940833847889 + 627189273993351994880472X +174238926938195499787264X2 −772 11 · 19(112) 23022526393975146777− 11207080634279747904X + 78906790753194116752X2 −2020 11 · 23(112) 485942503741395741 + 164603068156444320X + 212641567145618000X2 −1780 13 · 19(132) −7284723384049281 + 1837303480723680X + 4187652728072000X2 −955 72 −577517467114068471− 897234444403808940X + 76097089966140100X2 −1555 172(17 · 23) −169396006577729486341671 + 28230373334921475598740X +8279977244361140484100X2 Table 6.10: Some N for which ΨΓ+,N,t+10(X,X) vanishes at t + 10(CM(Λ)) and the irreducible polynomial of t+10(CM(Λ)) 220 Computational applications: kroneckerian polynomials and expansions of automorphic functions and forms The case D = 22 In this case, the orders whose corresponding points on the curve X(22, 1)+ are rational or quadratic are given in the following table: f 1 3 5 7 |D(K)| s.t. the order of conductor f gives rational CM points 3, 4, 11, 20, 67, 88, 132, 148, 163, 187, 232, 1012 3, 4, 11 4 |D(K)| s.t. the order of conductor f gives quadratic CM points 56, 91, 115, 136, 168, 235, 264, 267, 280, 308, 312, 328, 372, 388, 408, 427, 520, 532, 616, 627, 660, 708, 715, 760, 1243, 1320, 1507, 1540, 1672, 1848 20, 88 3, 11 3, 4 Now, following the same approach as in the two previous cases, we find suitable values of N for each of the different possible Λ giving rational points in X(22, 1)+ such that ΨΓ+,N,t+22(X,X) vanishes at t + 22(P ) for all P ∈ CM(Λ). These values of N together with the corresponding value of the Hauptmodul t+22 at CM(Λ) are given in Table 6.11. Note that the values at the elliptic points of X(22, 1)+ are already known, three of them by definition of t+22, and the other, t+22(CM(Z[ 1+ √−3 2 ])), was obtained during the computation of the automorphic derivative in Section 3.4.2, cf. Theorem 3.1. D(Λ) N t+22(CM(Λ)) −3 (3) 2716 −4 (5) 1 −11 (3) ∞ −88 (13) 0 −20 3, 5 516 −132 3(7) 2516 −4 · 32 5, 13 4916 −11 · 32 5(23) 4948 D(Λ) N t+22(CM(Λ)) −3 · 32 7, 13 7564 −187 7(17) − 1764 −4 · 52 13, 17 729784 −67 17, 19 1675784 −1012 17(23) − 342258624 −148 19(23) 18132704 −232 29(31) 23121421 −163 41 11882792416 Table 6.11: Some N for which ΨΓ+,N,t+22(X,X) vanishes at t + 22(CM(Λ)) and value of t+22(CM(Λ)) 6.2 Computation of the values of a Hauptmodul and its expansions at a given CM point 221 Similarly, the values at the quadratic CM points of the Hauptmodul t+22 together with values of N providing some ΨΓ+,N,t+22(X,X) having the corre- sponding value as a root are given in Table 6.12. D(Λ) N Irr(t+22(P ), X) for P ∈ CM(Λ) −264 3(17) 297− 544X + 256X2 −616 7, 29 −6561 + 800X + 1792X2 −308 7, 37 −175− 32X + 256X2 −56 7(3 · 5) 2023− 3936X + 1792X2 −660 13(3 · 5) 1225− 1685X + 784X2 −136 17, 19 12393− 6944X + 4352X2 −627 17(3 · 13) 893025− 813888X + 200704X2 −3 · 52 19, 31 2809 + 2896X + 4096X2 −715 19, 5 · 13 893025− 1689280X + 802816X2 −1672 19(72) 29452329− 42546464X + 15462656X2 −168 23, 3 · 7 133225− 135968X + 12544X2 −91 23, 29 35721− 30016X + 4096X2 −20 · 32 23, 5 · 29 6241− 10381X + 4624X2 −1540 23(5 · 7) 5267025− 17575840X + 12503296X2 −115 29, 41 164025− 327280X + 173056X2 −4 · 72 29, 37 6561− 167584X + 92416X2 −88 · 32 29(31) 7425− 17568X + 12544X2 −11 · 52 31, 3 · 23 14161− 16880X + 5120X2 −1243 31(17 · 19) −51480625 + 28113200X + 7614464X2 −280 37, 5 · 7 1225449− 3335584X + 2119936X2 −1320 37, 3 · 5 20638849− 12202784X + 2119936X2 −3 · 72 37, 3 · 13 2175625− 3349568X + 1183744X2 −1507 37(132) −332150625− 140259600X + 159509504X2 −312 41, 3 · 13 22325625− 34538400X + 12503296X2 −328 41(7 · 13) 7405625− 10400800X + 3789056X2 −1848 3 · 7 1586030625− 2350404000X + 765296896X2 −372 3 · 17, 3 · 29 893025 + 242163X + 26896X2 −408 3 · 17(3 · 37) 45630025− 84318368X + 62220544X2 −267 3 · 23(3 · 29) 583947225− 621513648X + 39337984X2 −708 3 · 31 9365400625− 11007447425X + 3148107664X2 −520 5 · 13, 5 · 31 393824025− 330322720X + 79995136X2 −235 5 · 13(5 · 23) 24950025− 5043040X + 3625216X2 −760 5 · 19(5 · 29) 3318336025− 25543986080X + 24131758336X2 −532 7 · 13(7 · 19) 34164025− 308664608X + 275294464X2 −427 7 · 17(7 · 31) 46831124025− 71808216928X + 26254369024X2 −388 7 · 19(7 · 23) 455625− 642025X + 29584X2 Table 6.12: Some N for which ΨΓ+,N,t+22(X,X) vanishes at t + 22(CM(Λ)) and the irreducible polynomial of t+22(CM(Λ)) 222 Computational applications: kroneckerian polynomials and expansions of automorphic functions and forms 6.3 Computation of the local parameters Given a CM point P ∈ H, we consider expansions around P with respect to the local parameter qP (z) = κP z − P z − P , for a suitable local constant κP ∈ C ∗, of the form f(qP ) = ￿ i≥n0 aiq i P . Moreover, given a Hauptmodul t of the corresponding curve we will choose κP such that, if P has order e, then ae = 1, if P is not a pole of t, and a−e = 1, if P is a pole of t. We will restrict to the cases D = 6, 10, 22 and will provide q-expansions around all the points where t+D admits a rational q-expansion, which coincides, as we have seen, with the set of points where t+D takes rational values and which has just been computed. The q-expansions for the case D = 6 can be found in Table 6.13, those for D = 10 in Table 6.14 and those for D = 22 in Table 6.15. Next we are going to deal with the problem of finding the local parameters which correspond to all the expansions we have given in this section. In order to do so, it will be useful to recall Corollary 5.2 which gives more information about the transcendental nature of the local constant κP corresponding to a CM point P . Apart from the values at a very limited set of points in the case D = 6 all of them have been computed numerically at high precision and then, after taking care of the transcendental part, the algebraic part is recognized. We do not know any way to certify that the given values are indeed exactly correct, but all of them match the real ones up to 150 decimal digits. Given a Hauptmodul t for the curve X(D, 1)W and a non-elliptic CM point P ∈ H for an order Λ in a quadratic imaginary field K of discriminant DK such that t(P ) ￿= ∞, we consider κP ∈ C∗ such that t(z) = ￿ n anqP (z) n with a1 = 1 as introduced above. In this situation, κP = (P − P )∂f ∂z (P ). In particular, if we know an expansion around a point Q, f = ￿ n bnqQ(z) n, which converges at P , we can obtain an approximation of kP as follows: κP = κQ (P − P )(Q−Q) (P −Q)2 ∂f ∂qQ (qQ(P )). Note that it does not matter if Q is elliptic or not. 6.3 Computation of the local parameters 223 D(Λ) t+6 (q), P ∈ CM(Λ) −3 1q6 + 167280 + 854531281280q6 + · · · −4 1 + q4 + 103270q8 + · · · −24 q2 − 103216q4 + 141191933120q6 + · · · −40 2312125 + q + 2228756741792q2 + . . . −84 − 16927 + q − 23103264992q2 + . . . −19 32111024 + q + 5726722340819q2 + . . . −120 57763375 + q + 3614737555472704q2 + . . . −51 − 13771024 + q − 11031043306177q2 + · · · −52 687715625 + q − 14171875160426656q2 + · · · −132 1368915625 + q − 573859375212015232q2 + · · · −168 − 70178415625 + q − 260704843752013864630624q2 + · · · −3 · 52 152881138240 + q + 83074752002238330721q2 + · · · −43 2125098716000000 + q + 4829272400000037196218824723q2 + · · · −88 1554588820796875 + q − 122816340453125108841674388608q2 + · · · −123 − 29690072116000000 + q − 286908763600000092900449666319841q2 + · · · −228 6686332911390625 + q + 337993530004687529672717264572928q2 + · · · −312 2700874238427680640625 + q − 109649727388722485937572588505997726986176 q2 + · · · −3 · 72 − 10731520813024000000 + q − 42277252467240000004396867281898630561q2 + · · · −4 · 52 4218505211771561 + q + 1167053473908421472561408365701760q2 + · · · −67 779037006671024000000 + q + 155571074716160000001996404396043509414963q2 + · · · −372 − 455413074649747377296875 + q − 27446030396521353493593754382131690031662847160608q2 + · · · −408 − 3240860943673655962140625 + q − 42523502768378847023906254208526483126612133037334784q2 + · · · −148 69630712957377149515625 + q + 5165633692245434557812557100675939882223384736q2 + · · · −267 − 57666817144887211814078464000000 + q − 704249947889241377715942400000043715831103215160241138980215841q2 + · · · −232 6643227848345223256413239012828125 + q + 2023469428190655366812375638296875887450160332267339060091949542432 q2 + · · · −708 71475755554842930369224337327397603890625 + q + 2151241177917897119730143272404106057812587407170742097947046972030144251075315712q2 + · · · −163 699690239451360705067684178814003344000000 + q + 603216261026081587096390861064693960000003617730995318246123710922456278437158163 q2 + · · · Table 6.13: First terms of the q-expansion of t+6 at the CM points by the order of discriminant D(Λ) 224 Computational applications: kroneckerian polynomials and expansions of automorphic functions and forms D(Λ) t+10(q) for P ∈ CM(Λ) −3 272 + q3 + 34675q6 + · · · −8 1 + q2 + 43150q4 + · · · −20 1 q2 + 356 + 611 48 q 2 + · · · −40 q2 − 5051458q4 + · · · −35 − 52 + q − 101560q2 + · · · −120 545 + q − 81510584q2 + · · · −52 729104 + q + 9854455625q2 + · · · −280 − 6561845 + q − 18136085435258697238184q2 + · · · −3 · 32 867242 + q + 3614272890000q2 + · · · −115 − 3645242 + q − 723196431813518720q2 + · · · −20 · 32 289120 + q + 1562931065007371q2 + · · · −43 31347722 + q + 31903411115360030000q2 + · · · −88 7293179 + q + 664789301893025000q2 + · · · −8 · 32 3844169 + q + 3841577877063350000q2 + · · · −340 − 1822522984 + q − 42012284265389137830982175q2 + · · · −520 12131289874225 + q + 1215211476909752492513322510436874687776q2 + · · · −235 − 306544587362 + q − 14365338484281983759727175345071920q2 + · · · −67 439587232562 + q + 67591577001553182010997350000q2 + · · · −148 5517801342176 + q + 579006012097386438010789573631875q2 + · · · −760 294523290531898939 + q − 18633557057828302541725631142644863435237646735755160q2 + · · · −232 106571781171374281 + q − 50792550840646244536283180642025945017051030000q2 + · · · −163 18300320498715685392962 + q − 90401671128930817031761845947652077042863923852878502800000q2 + · · · Table 6.14: First terms of the q-expansion of t+10 at the CM points by the order of discriminant D(Λ) 6.3 Computation of the local parameters 225 D(Λ) t+22(q) for P ∈ CM(Λ) −3 2716 + q3 + 232297q6 + · · · −4 1 + q4 − 91330q8 + · · · −11 1 q2 + 2324 + 559 3840q 2 + · · · −88 q2 − 33895832q4 + · · · −20 516 + q + 855q2 + · · · −132 2516 + q − 424225q2 + · · · −4 · 32 4916 + q + 6641617q2 + · · · −11 · 32 4948 + q + 168398 q2 + · · · −3 · 32 7564 + q + 432275q2 + · · · −187 − 1764 + q − 2430419125q2 + · · · −4 · 52 729784 + q − 20501640095 q2 + · · · −67 1675784 + q + 336973331650q2 + · · · −1012 − 342258624 + q − 188039740319211922213277575q2 + · · · −148 18132704 + q − 6015994467307625q2 + · · · −232 23121421 + q − 1636930897343332000 q2 + · · · −163 11882792416 + q + 216379056448392292487125q2 + · · · Table 6.15: First terms of the q-expansion of t+22 at the CM points by the order of discriminant D(Λ) 226 Computational applications: kroneckerian polynomials and expansions of automorphic functions and forms Therefore, we will obtain the values of κP for all the points P above, from the values at a more restricted set of CM points, which we will take as the set of elliptic points. In order to obtain approximated values for the local constants at the set of elliptic points, we will use Section 3.3.1. In particular, if we assume that t is a Hauptmodul satisfying the differential equation Da(t, z) +R(f(z)) = 0 and that P is an elliptic point such that t(P ) = a ￿= ∞, in Section 3.3.1, we have seen that κP Q− P Q− P qP (Q) = u1(t(Q)) u2(t(Q)) , where u1, u2 are solutions of the linear differential equation u￿￿(w)+R(w)/4 = 0 with initial conditions u1(a) = 0, u￿1(a) = 1, u2(a) = 1, u￿2(a) = 0 and Q lies in a suitable neighborhood of P . Note that assuming that we know the values of t at the elliptic points is no restriction after assuming that we know R, since these values are the zeros of the denominator of R. Moreover, we have a method to compute the value of t at a given CM point, in particular t(Q) for a suitable Q. Finally, for the case t(P ) = ∞, we consider the Hauptmodul 1/t, which reduces to the situation above. Let P ∈ H be a non-elliptic CM point by an order Λ in a quadratic field K of discriminant DK where t+D takes a real value. Assume that we have t+D(qP (z)) = ￿ n anq n P , with a1 = 1. Then, since t + D is defined over Q, it follows from [Shi75b] that t+D(s(z)) = t + D(z), where s is the symmetry attached to an element of norm −1 in O(D, 1). The set of fixed points of this symmetry coincides with the pictured lines in the fundamental domains of ￿Γ(D, 1),WD￿ given in Figures 3.5, 3.6 and 3.4 and corresponds to the set of points where t+D takes real values. In particular, the point P must lie in one of these hyperbolic lines (and only one if the point is not elliptic), assume that it is an euclidian circle of center a. Note as well that a ∈ Q ∪ {∞} since this is the set of fixed points of a symmetry given by a matrix with algebraic coefficients. Then, the directional derivative along this line must be real and this allows us to break down κP a little more. More precisely, we fix a tangent vector to the hyperbolic line of center a through P , which for convenience we are going to choose as follows: iva,P , where va,P = P/a− 1 if a ￿= 0 and va,P = P/Re(P ) if a = 0. Then, the directional derivative along iva,P at P is ￿ ∂f ∂z ￿ (P ) · iva,P ∈ R, 6.3 Computation of the local parameters 227 which can be computed as￿ ∂f ∂z ￿ (P ) · iva,P = ￿ ∂f ∂qP ￿ (0)q￿P (P ) · iva,P = κP va,P 2Im(P ) . In particular κP ∈ va,PR. Therefore, since a is algebraic, we have κP va,PπDK ∈ R ∩Q, where πDK is the constant introduced in Corollary 5.2. Note that if we change P ∈ H by Q = γ(P ) for a certain γ ∈ Γ+, the expansion at Q, ￿ n anq n Q, has the same coefficients as the expansion at P but the local constant at Q is not exactly the same: if we write γ = ￿ a b c d ￿ , then we have that κQ = a−cQa−cQκP . For this reason we are going to choose representatives of all the points above in the fundamental domain D(D, 1)+ and for each of these points we will give the corresponding local constant. More precisely, in Tables 6.16, 6.17, 6.18, for the non-elliptic rational CM points, we provide a representative point P ∈ D(D, 1)+, the corresponding values of a and κP va,PπDK , which determine the values of κP corresponding to the expansions for t+D given above. Similarly, for an elliptic CM point of order e we provide as well a representative point P ∈ D(D, 1)+ and the value of ￿ κP πDK ￿e . It is worth noting that from one of these expansions at a point P we can obtain expansions for any of the functions uD,i and tD at those points mapped to P , since these functions are algebraically related to t+D by means of the polynomials given in Theorem 2.4. Finally, let us remark that the only three values which are unconditionally correct are the ones corresponding to the elliptic points in the case D = 6. In this case there are only three elliptic cycles and this allows to rewrite the function u1/u2 in terms of hypergeometric series. More details can be found in [BT07b], where these three local constants at the elliptic points together with the corresponding expansions are computed using this fact. 228 Computational applications: kroneckerian polynomials and expansions of automorphic functions and forms DK f(Λ) P e(P ) (κP /πDK ) e(P ) −3 1 Q+6,2 = −1+i1+√3 6 2i729 −4 1 Q+6,1 = i 4 − 364 −24 1 Q+6,4 = i √ 2 1+ √ 3 2 14 DK f(Λ) P a κP va,P πDK −40 1 − √ 3i √ 10 4+ √ 3 1 8262 √ 3 625 −84 1 −−1+i √ 7 2(1+ √ 3) 0 91 √ 6 81 −19 1 − √ 3+i √ 19 5+ √ 3 1 3159 √ 3 2048 −120 1 −1+i √ 10 1+2 √ 3 1 13034 √ 6 50625 −51 1 −1+i √ 17 3(1+ √ 3) 0 10292048 −52 1 i √ 13 4+ √ 3 ∞ − 22356 √ 3 78125 −132 1 i √ 33 6+ √ 3 ∞ − 10296 √ 2 78125 −168 1 −2+i √ 14 3(1+ √ 3) 0 2124276 √ 3 78125 −3 5 −1+5i 1+3 √ 3 1 520421 276480 √ 355/6 −43 1 − √ 3+i √ 43 7+ √ 3 1 58594347 √ 3 160000000 −88 1 i √ 22 5+ √ 3 ∞ − 232377012 √ 6 1143828125 −123 1 −3+i √ 41 5(1+ √ 3) 0 18992838501800000000 −228 1 −2+i √ 19 2+3 √ 3 1 852632144 √ 2 170859375 −312 1 i √ 78 9+ √ 3 ∞ − 109824062796 √ 3 1522435234375 −3 7 −1+7i 5(1+ √ 3) 0 530508320243 1058400000000 √ 371/6 −4 5 −2 √ 3+5i 7+2 √ 3 1 13695240888 √ 3 1948717151/4 −67 1 −3 √ 3+i √ 67 11+3 √ 3 1 3782858460381 √ 3 20480000000 −372 1 −1+i √ 31 4(1+ √ 3) 0 49217478940721 41105751328125 √ 6 −408 1 −4+i √ 34 5(1+ √ 3) 0 154852225464272336 √ 2 71351729296875 −148 1 i √ 37 7+2 √ 3 ∞ − 19250322046956 √ 3 32057708828125 −267 1 −3+i √ 89 7(1+ √ 3) 0 2802694559631731037399097262080000000 −232 1 − √ 3+i √ 58 8+ √ 3 1 138679998463115343234 √ 3 188138152107781796875 −708 1 i √ 59 4+5 √ 3 ∞ − 1513133032990653000134208 √ 2 552991512035093590390625 −163 1 − √ 3+i √ 163 13+ √ 3 1 389105683427842995320343 √ 3 1279414382186253280000000 Table 6.16: Local constants around the rational CM points in X(6, 1)+ normal- ized for the Hauptmodul t+6 6.3 Computation of the local parameters 229 DK f(Λ) P e(P ) (κP /πDK ) e(P ) −3 1 Q+10,2 = 2−i √ 6 5(−2+√2) 3 1 10 (−14 + 3i √ 6) −8 1 Q+10,3 = −1+2i5 2 316 − i4 −20 1 Q+10,1 = i(1+ √ 2)√ 5 2 − 1100 −40 1 Q+10,6 = i√5 2 − 81400 DK f(Λ) P a κP va,P πDK −35 1 i(1+2 √ 2)√ 35 ∞ 4 √ 10 5 −120 1 2i √ 3−√5 5 √ 5−2√10 −1 − 18950√2 −52 1 3−i √ 26 5(−3+√2) −1 − 303755408√2 −280 1 i(3+ √ 2)√ 35 ∞ 2028807 54925 √ 10 −3 3 2 √ 2−3i√3 5(1−2√2) −1 − 4250 √ 2 1331 −115 1 i(3+4 √ 2)√ 115 ∞ 50544 √ 10 6655 −20 3 5−3i √ 10 5(−5+√2) −1 − 267417200√6 −43 1 √ 2−i√43 15(1−√2) 0 113400 6859 −88 1 115 (−1 + 2i √ 11) 0 − 70875 594473 √ 2 −8 3 −1+3i 5(2−√2) 0 27125 2197 √ 6 −340 1 i(1+3 √ 2)√ 85 ∞ 26095041 10158928 √ 2 −520 1 i(5i+ √ 65) 15(2−√2) 0 2891123361 817400375 √ 5 −235 1 i(5+6 √ 2)√ 235 ∞ 2109899988 √ 10 45646645 −67 1 3 √ 2−i√67 5(1−3√2) −1 − 82923750 √ 2 39651821 −148 1 2 √ 2−i√37 15(1−√2) 0 1939862925 430457408 −760 1 145 (−5 + 2i √ 95) 0 − 53379126511389 53471268438731 √ 2 −232 1 125 (−3 + 2i √ 29) 0 − 14570181791552243460712571 −163 1 i(12i+ √ 326) 5(12−5√2) −1 − 14925693317535000 √ 2 694541357660879 Table 6.17: Local constants around the rational CM points in X(10, 1)+ nor- malized for the Hauptmodul t+10 230 Computational applications: kroneckerian polynomials and expansions of automorphic functions and forms DK f(Λ) P e(P ) (κP /πDK ) e(P ) −3 1 Q+22,2 = i √ 3−√11 5+ √ 11 3 (P−1) 3 32 √ 11 −4 1 Q+22,1 = i 4 − 1704 −11 1 Q+22,3 = −1+i3+√11 2 − 8i121 −88 1 Q+22,6 = i √ 2 3+ √ 11 2 81484 DK f(Λ) P a κP va,P πDK −20 1 i √ 5 4+ √ 11 ∞ − √ 11 16 −132 1 −2+i √ 3 2+ √ 11 1 15 176 √ 2 −4 3 3i− √ 11 8+2 √ 11 3−√11 731/4 √ 11 8 √ 2 −11 3 −1+3i 1+ √ 11 1 7 198 √ 3 −3 3 3i √ 3−√11 7+ √ 11 1 5 √ 11 128 −187 1 −1+i √ 17 9+3 √ 11 0 2251408 −4 5 5i 6+ √ 11 ∞ − 243 √ 11 548851/4 −67 1 −3 √ 11+i √ 67 21+5 √ 11 3−√11 135 √ 11 343 −1012 1 −3+i √ 23 12+4 √ 11 0 289854855 2656192 √ 22 −148 1 i √ 37 9+2 √ 11 ∞ − 4725 √ 11 35152 −232 1 −3 √ 11+i √ 58 16+3 √ 11 1 22950 √ 11 288463 −163 1 −3 √ 11+i √ 163 19+3 √ 11 1 2083725 √ 11 7023616 Table 6.18: Local constants around the rational CM points in X(22, 1)+ nor- malized for the Hauptmodul t+22 6.4 Automorphic forms, Hecke operators and q-expansions 231 6.4 Automorphic forms, Hecke operators and q- expansions In this section we will consider the graded Q-algebra of cusp forms S∗(Γ(D,N)), which we introduced in Section 1.3, for D = 6, N = 1, 5, 7, 11, D = 10, N = 1, 3, 7, 9 and D = 22, N = 1, 3, 5, 7. In all these cases we will give a set of generating forms for this Q-algebra. In order to do so we will use the Hauptmoduln t+D, which we introduced in Theorem 2.4 together with the properties given in Theorem 3.1 and in Section 6.2.1, and t+D,N , N > 1, which we computed and studied in Chapter 4. The study of the Hecke operators will be carried out, when necessary, through a construction similar to that of the kroneckerian polynomials. The computation of q-expansions for automorphic forms around a CM point will then follow from the knowledge of the q-expansions for t+D and t + D,N around this point. The ring of automorphic forms corresponding toD = 6, N = 1 has been studied in [BG08] and q-expansions around the elliptic points in this case can be found in [BG12], where they are computed following the approach given in [Zag08] and a recursion formula for the coefficients is provided. Given an automorphic form of weight 2t (or a meromorphic form which is analytic at P ), we will consider expansions around a CM point P of the form f = 1 (2πi)t ￿ (q − κ)2 κ(P − P ) ￿t￿ n≥0 can qn n! , where q = κ z−P z−P ,κ ∈ C∗, is a suitable local parameter and c ∈ C∗ is a normaliz- ing constant. Note that these are just a normalized version of those considered in Chapter 5 and coincide as well, up to normalization, with those considered in [Zag08] and [BG12]. Example. If we consider the modular curve of level 1, X0(1), its function field is generated by the classical j-function, which is usually identified with its Fourier expansion j(q∞) = 1 q∞ + 744 + 196884q∞ + 21493760q2∞ + . . . where q∞(z) = e2πiz for z ∈ H. Moreover, the algebra of modular forms is generated by E4, E6,∆, where E4 and E6 denote the normalized Eisenstein series of weights 4 and 6, respectively, and ∆, the modular discriminant. 232 Computational applications: kroneckerian polynomials and expansions of automorphic functions and forms Then, E4 = ￿ 1 2πi dj dz ￿2 1 j2 − 1728j , E6 = − ￿ 1 2πi dj dz ￿3 1 j2(j − 1728) , ∆ = ￿ 1 2πi dj dz ￿6 1 j4(j − 1728)3 . This can be easily seen comparing the first terms of the Fourier expansions and checking that the divisors of both sides agree. To do that, we only need to remember that the only multiple points of j in H are: ρ = e2πi/6, where j(ρ) = 0 with multiplicity 3, and i, where j(i) = 1728 with multiplicity 2. Note also that 12πi d dz is nothing else than the differential operator more usually written as q∞ ddq∞ . Therefore, from the q-expansion of j around a CM point (or the Fourier expansion at ∞) we can easily compute q-expansions (or Fourier expansions) for these forms. Expansions for j around certain CM points are given in [BT07a] and expansions around other CM points can be computed using the procedure described in Section 6.2. Expansions around the elliptic point i, of order 2, and ρ, of order 3, and also around √−6, a CM point by the ring of integers of Q(√−6) can be found in Tables 6.19, 6.20, 6.21. The values of the corresponding local constants for the CM points by the rings of integers of Q(i), Q( √−3) and Q(√−6) are deduced from the ones computed in [BT07a]. c 0 1 2 3 4 5 6 7 8 9 10 11 j 576 3 0 36 0 1104 0 49816 0 2973824 0 227654496 0 E4 8 3 0 10 0 30 0 360 0 2280 0 66960 0 E6 −96 0 3 0 14 0 168 0 1764 0 43848 0 864864 ∆ 8 1 0 −2 0 6 0 48 0 −1584 0 6048 0 Table 6.19: Coefficients of the expansions around i with κi = 12√2π−4 6.4 Automorphic forms, Hecke operators and q-expansions 233 c 0 1 2 3 4 5 6 7 8 9 10 11 12 13 j 1728 0 0 0 8 0 0 −460 0 0 53334 0 0 −10612635 0 E4 −6 0 2 0 0 −5 0 0 30 0 0 −600 0 0 15360 E6 18 2 0 0 −7 0 0 63 0 0 −1428 0 0 52248 0 ∆ −3/4 1 0 0 1 0 0 8 0 0 120 0 0 372 0 Table 6.20: Coefficients of the expansions around ρ with κρ = − 3 √ 6 3 π−3 c 0 j 1728 1399 + 988 √ 2 E4 −18 ￿−1954045 + 1381718√2￿ 68 E6 216 ￿−5463983723 + 3863619942√2￿ 92 ∆ −108 ￿−83452309924610425346859 + 59009694253373452368026√2￿ 2 1 2 j 79764 −204 ￿−15774652 + 11145963√2￿ E4 −40 ￿−8971 + 6339√2￿ −20 ￿−879357913 + 621799902√2￿ E6 −84 ￿−25090 + 17737√2￿ −42 ￿−663024313 + 468828932√2￿ ∆ 12 ￿−1147 + 809√2￿ 6 ￿−32318107 + 22852414√2￿ 3 4 j −6 ￿−41259885695815 + 29175135775248√2￿ −36 ￿−648890835855607688 + 458835110231329411√2￿ E4 −60 ￿−1693258199391 + 1197314355013√2￿ −30 ￿−389630223646822311 + 275510173295898148√2￿ E6 −168 ￿−11737526010111 + 8299684235999√2￿ −756 ￿−65827732575595575 + 46547236094338036√2￿ ∆ −6 ￿−1643944362399 + 1162444207295√2￿ −153 ￿−328527266679847 + 232303858073740√2￿ Table 6.21: Coefficients of the expansions around P = √−6 with κP = (2765+1956 √ 2) 79764 π−24 Let us now move on to the non-modular case. Apart from the case D = 6, N = 1 where the ring of automorphic forms is studied in [BG08], we have a priori no known generators of the Q-algebra of automorphic forms. However, in all the cases we are going to work with, we have that X(D,N)+ has genus 0 and we already know a rational Hauptmodul for this curve, t+D,N . Therefore, the following result will be useful to explicitly determine the cusp forms of a certain weight. Proposition 6.5. Let X(D,N) be a Shimura curve attached to an Eichler order of level N in a non-split indefinite quaternion algebra of discriminant D such that g(X(D,N)+) = 0 (cf. Proposition 4.1). Then, given a positive integer t, S2t(Γ(D,N))Q = S2t(Γ(D,N) +)Q ⊕ ￿ W S2t(￿Γ(D,N),W ￿)−Q , 234 Computational applications: kroneckerian polynomials and expansions of automorphic functions and forms where W runs over the set of subgroups of WDN of index 2 and S2t(￿Γ(D,N),W ￿)−Q ⊂ S2t(￿Γ(D,N),W ￿)Q is the eigenspace of eigenvalue −1 by the non-trivial element in WDN/W . Let t+D,N be a Hauptmodul for X(D,N)+ defined over Q and fix W ⊂ WDN a subgroup of index at most 2. Denote by {Pi}i ⊂ Γ(D,N)+\H the set of points ramified by the quotient map X(D,N)W (C)→ X(D,N)+(C) and by {Qi} the remaining elliptic points in X(D,N)+(C). Then, if W has index 2, we consider uW ∈ Q(X(D,N)W ) such that u2W = uW,0 ￿ i (t+D,N − t+D,N (Pi)), where the product runs over the set of Pi where t+D,N is finite and uW,0 ∈ Z is a squarefree integer. In the case W = WDN , we set uWDN = 1. Then, either S2t(￿Γ(D,N),W ￿)−Q , if W ⊂ WDN has index 2, or S2t(Γ(D,N)+)Q, if W =WDN , can be written as D(t+D,N ) t uW Φ(t+D,N ) r￿ i=1 (t+D,N − t+D,N (Pi))αi s￿ i=1 (t+D,N − t+D,N (Qi))βi , where the products are restricted to the points where t+D,N is finite, and αi = ￿ t(e(Pi)− 1) e(Pi) − 1 2 ￿ , βi = ￿ t(e(Qi)− 1) e(Qi) ￿ , where e(P ) denotes the elliptic order of P ∈ X(D,N)+(C) and Φ(X) ∈ Q[X] is a non-zero polynomial of degree at most g(X(D,N)W ) + 1− 2t+ ￿ i αi + ￿ i βi. Remark. We could use the above proposition to recover the formulas for the dimension of Sk(Γ(D,N)) given in Theorem 1.20 in this case. 6.4 Automorphic forms, Hecke operators and q-expansions 235 The functions uW for all the cases we will deal with can be found in Theo- rem 2.4, if N = 1, and in Chapter 4, if N > 1. The expression we obtain in the case W =WD,N is the classical description of the ring of automorphic forms for genus 0 curves in terms of a Hauptmodul, which is classically found in the setting of modular forms (cf. [Sch74]), but can also be found in the case of Shimura curves, cf. [Yan13]. Proof. The space of meromorphic automorphic forms of weight 2t with re- spect to Γ(D,N) which are defined over Q, A2t(Γ(D,N))Q, can be written as D(t+D,N )tQ(X(D,N)). But not only this, since S2t(Γ(D,N))Q has a ba- sis of eigenforms, and the action of the Hecke operators is compatible with the action of the Atkin-Lehner involutions, we can decompose a cusp form into a sum of forms in S2t(Γ(D,N)+)Q and S2t(￿Γ(D,N),W ￿)−Q for all the W ⊂WDN such that [WDN :W ] = 2. Therefore, to write down the elements in S2t(Γ(D,N))Q it suffices to determine the holomorphic elements in the various spaces D(t+D,N )tQ(X(D,N)W ). Observe now that Q(X(D,N)) = Q(t + D,N ) and, since Q(X(D,N)W ) is a quadratic extension of Q(t+D,N ), Q(X(D,N)W ) = Q(t+D,N , uW ), for a certain uW ∈ Q(X(D,N)W ) such u2W = uW,0 ￿ i(t + D,N − t+D,N (Pi)), where uW,0 ∈ Z is a suitable squarefree integer and {Pi}i ⊂ X(D,N)+(C) denotes the set of points where t+D,N is finite and which ramify in the quotient map X(D,N)W (C)→ X(D,N)+(C). The number of points of X(D,N)+(C) which are ramified by this quotient map can be computed using the Riemann-Hurwitz formula and it is equal to r = 2g(X(D,N)W ) + 2. In particular, the sub- space S2t(￿Γ(D,N),W ￿)−Q coincides with the set of holomorphic elements in D(t+D,N ) t uW Q(t+D,N ). Now, an element of this last set can be written as D(t+D,N ) t uW Φ0(t + D,N ) Φ1(t + D,N ) , for some coprime polynomials Φ0(X),Φ1(X) ∈ Q[X], and therefore the set of zeros of Φ1(t+D,N ) must be contained in the set of zeros of D(t+D,N ) t uW and since this is a subset of {Pi, Qj}, we obtain that it can be written as F ({αi}, {βi},Φ) = D(t+D,N ) t uW Φ(t+D,N )￿ i(t + D,N − t+D,N (Pi))αi ￿ i(t + D,N − t+D,N (Qi))βi , 236 Computational applications: kroneckerian polynomials and expansions of automorphic functions and forms for some positive integers αi,βi. In particular, we already have that this expres- sion is holomorphic away from {Pi, Qj} and the pole of t+D,N . Thus, we only have to find out which conditions have to be imposed on αi,βi and degΦ so that it is holomorphic at these points as well. If we compute ordPi F ({αi}, {βi},Φ), we obtaine(Pi) ￿ t e(Pi)−1e(Pi) − 12 − αi ￿ , if t+D,N (Pi) ￿=∞, e(Pi) ￿ −t e(Pi)+1e(Pi) − degΦ+ r−12 + ￿ j ￿=i αj + ￿ j βj ￿ , if t+D,N (Pi) =∞. Similarly, the order of the function F ({αi}, {βi},Φ) at any of the two points in X(D,N)W (C) which map to Qi is equal toe(Qi) ￿ t e(Qi)−1e(Qi) − βi ￿ , if t+D,N (Qi) ￿=∞, e(Qi) ￿ −t e(Qi)+1e(Qi) − degΦ+ r2 + ￿ j αj + ￿ j ￿=i βj ￿ , if t+D,N (Qi) =∞. Otherwise, if the pole P of t+D,N is not in {Pi, Qi}, ordP F ({αi}, {βi},Φ) = −2t− degΦ+ r 2 + ￿ j αj + ￿ j βj . Therefore, we obtain that αi ≤ ￿ t(e(Pi)− 1) e(Pi) − 1 2 ￿ , βi ≤ ￿ t(e(Qi)− 1) e(Qi) ￿ , and degΦ ≤ g + 1− 2t+ ￿ j αj + ￿ j βj . Finally, if we no longer assume that Φ(t+D,N ) is coprime to the denomina- tor, multiplying the numerator and the denominator by the necessary factors￿ i(t + D,N − t+D,N (Pi)), ￿ i(t + D,N − t+D,N (Qi)), we can always assume that αi = ￿ t(e(Pi)− 1) e(Pi) − 1 2 ￿ , βi = ￿ t(e(Qi)− 1) e(Qi) ￿ , which is the expression given in the statement of the proposition. Using this result we can explicitly compute the space S2t(Γ(D,N))Q for a fixed positive integer t. However, we are interested in computing a set of 6.4 Automorphic forms, Hecke operators and q-expansions 237 generators of S∗(Γ(D,N))Q, the algebra of cusp forms. In order to do so, we will compute the elements of this algebra up to a certain weight, and then we need to check whether they generate the full algebra or not. We could use the explicit descriptions given in the previous proposition to prove that the elements up to a certain weight suffice to generate the algebra. However, it will be more convenient to use the following criterion. Proposition 6.6. Let f1, . . . , fr ∈ S∗(Γ(D,N))Q, r ≥ 2, be forms of weights k1, . . . , kr ≥ 2. Consider the graded polynomial ring Q[X1, . . . , Xr] with weights (k1, . . . , kr) and the natural surjective morphism φ : Q[X1, . . . , Xr]→ Q[f1, . . . , fr] ⊂ S∗(Γ(D,N)) such that φ(Xi) = fi. Let I ⊂ Q[X1, . . . , Xn] be an ideal and assume that (a) fk21 f −k1 2 /∈ Q, (b) I ⊂ kerφ, (c) dim (Q[X1, . . . , Xr]/I) = 2, (d) I is prime. Then, I = kerφ. In particular, Q[f1, . . . , fr] = S∗(Γ(D,N))Q if and only if both rings have the same Hilbert series, in which case Q[X1, . . . , Xr]/I gives a presentation of S∗(Γ(D,N))Q. Proof. (a) is equivalent to fk21 and f k1 2 linearly independent over Q. Let us now see that two linearly independent forms g1, g2 of the same weight, are algebraically independent. Given a polynomial P (x, y) ∈ Q[x, y] such that P (g1, g2) = 0, if we write P (X,Y ) = P1(X,Y ) + · · · + Ps(X,Y ) as a sum of homogenous polynomials of distinct degrees, the transformation law for au- tomorphic forms, yields that Pi(g1, g2) = 0 for all i. In particular, we can assume that P is homogeneous. Now, it follows that P (g1g−12 , 1) = 0 which implies that g1g−12 is constant and therefore it lies in Q. We have proved indeed that Q(f1, . . . , fr) is an extension of transcendence degree 1 over a non-constant subextension of Q(X(D,N)) which also has transcendence degree 1. In partic- ular, the Krull dimension dimQ[f1, . . . , fr] is 2. Now, by (b), there exists an exhaustive morphism of Q[X1, . . . , Xr]/I into Q[f1, . . . , fr]. Since Q[f1, . . . , fr] is a domain, because it is a subalgebra of S∗(Γ(D,N))Q, we obtain that kerφ is a prime ideal containing I. In particular, 238 Computational applications: kroneckerian polynomials and expansions of automorphic functions and forms it follows from (c) and (d) that, if I ￿ kerφ, then dim(Q[X1, . . . , Xr]/ kerφ) < 2 and we reach a contradiction, since it surjects into Q[f1, . . . , fr] which has Krull dimension 2. The final claim follows from the fact that, given an inclusion of graded rational algebras B ⊂ A, then B = A if and only if dimQBk = dimQAk for all k, which is equivalent to the equality between their Hilbert series, i.e. the generating functions of the sequences (dimQBk)k, (dimQAk)k. More information on the notions of dimension theory of K-algebras and Hilbert series involved in the proof can be found, for example, in [ZS60] or [Mat87]. Finally, once we have a set of generators of the algebra, we are interested in studying how the Hecke operators act on these elements. For this purpose, given a cusp form of weight 2t, f ∈ S2t(Γ(D,N))Q, and a positive integer m coprime to DN , we consider the polynomial Φm,f (X) = ￿ ω∈O(D,N)1\O(D,N)￿m ￿ X −mtf |2tω ￿ , which is analogous to the construction of kroneckerian polynomials we consid- ered for functions in this setting of forms. Proposition 6.7. (1) For f ∈ S2t(Γ(D,N))Q, then Φm,f (X) ∈ S∗(Γ(D,N))Q[X]. More precisely, it is a polynomial in X of degree ψ(m), Φm,f (X) = ψ(m)￿ i=0 ciX ψ(m)−i, such that ci ∈ S2ti(Γ(D,N))Q. (2) The coefficients of the polynomial Φm,f (X) can be expressed in terms of the Hecke operators acting on the powers of f as follows: ci = ￿ j1+2j2+···+ψ(m)jψ(m)=i j1,...,jψ(m)≥0 ￿−1 m ￿j1+···+jψ(m) (Tmf)j1 · · · (Tmfψ(m))jψ(m) j1!j2! · · · jψ(m)!1j12j2 · · ·ψ(m)jψ(m) . In particular, c1 = − 1 m Tmf. 6.4 Automorphic forms, Hecke operators and q-expansions 239 Proof. The degree of Φm,f (X) follows from the computation of the cardinal of O(D,N)1\O(D,N)￿m in Proposition 2.1. The rest of the assertions in the proposition are a consequence of the formula given in (2), which follows from the expression of the elementary symmetric polynomials in terms of power sums, usually known as Newton polynomials. If we assume that g(X(D,N)+) = 0, we can say something more about this polynomial. Proposition 6.8. Under the assumptions of Proposition 6.5 and using the same notations, let f ∈ S2t(Γ(D,N)W )−Q or f ∈ S2t(Γ(D,N)+Q ). Then if m = p is prime, Φm,f (X) is either irreducible over the fraction field of S2t(Γ(D,N))Q or is the power of a factor of degree one. Proof. By Proposition 2.1, we have that the group Γ(D,N) acts transitively on the set {f |2tω}ω∈O(D,N)1\O(D,N)￿m . Therefore, if all the elements in this set are different, as occurrs in the case of the kroneckerian polynomials, this polynomial is irreducible over the fraction field of S2t(Γ(D,N))Q. Otherwise, we are going to see that this set, since m is prime, consists of a single element. In order to do so, let us write f = D(t+D,N )tg for some g ∈ Q(X(D,N)W ). Then, if we had f |2tω = f |2tω￿, for some ω,ω￿ ∈ O(D,N)1\O(D,N)￿m corresponding to different classes, dividing by D(t+D,N )t, we would have that D(t+D,N ◦ ω)t D(t+D,N ) t g ◦ ω = D(t + D,N ◦ ω￿)t D(t+D,N ) t g ◦ ω￿. Now we observe that D(t+D,N ◦ ω)t D(t+D,N ) t ∈ Q(t+D,N , t+D,N ◦ ω) = A0(ω−1Γ(D,N)ω ∩ Γ(D,N))Q, which can be explicitly written down by differentiating the corresponding kro- neckerian polynomial. Since the same holds replacing ω by ω￿, we obtain D(t+D,N ◦ ω)t D(t+D,N ) t (g◦ω) ∈ A0(ω−1Γ(D,N)ω∩Γ(D,N))Q∩A0(ω￿−1Γ(D,N)ω￿∩Γ(D,N))Q, and since ω and ω￿ correspond to different classes, this yields D(t+D,N ◦ ω)t D(t+D,N ) t (g ◦ ω) ∈ A0(Γ(D,N))Q. 240 Computational applications: kroneckerian polynomials and expansions of automorphic functions and forms Therefore, f |2tω ∈ S2t(Γ(D,N))Q and, by the transitivity of the action, all the elements are equal. 6.4.1 Cusp forms attached to Shimura curves of discrimi- nant 6 We will begin by providing a set of generators of theQ-algebra of cusp forms with respect to Γ(6, 1). In order to do so, recall the automorphic functions for Γ(6, 1) introduced in Theorem 2.4, which coincide with those given in [BT07b], and further studied in Theorem 3.1. In particular, the function t6 is a Hauptmodul over the fieldQ( √−3, i), i.e.Q(√−3, i)(X(6, 1)) = Q(√−3, i)(t6), and it satisfies the differential equation Da(t6, z) + 27t46 + 74t 2 6 + 27 36t26(t 2 6 − 1)2 = 0, together with the functions u6,1 = √−3 t 2 6 + 1 2t6 , u6,2 = √ 3 t26 − 1 2t6 , u6,3 = i t26 + 1 t26 − 1 , which lie in Q(X(6, 1)) and provide Hauptmoduln over Q for the Atkin-Lehner quotients. We also have a Hauptmodul for X(6, 1)+ given by t+6 = ￿ t26 + 1 t26 − 1 ￿2 , which satisfies the differential equation Da(t+6 , z) +R(t + 6 ) = 0, where R(t) = 108− 113t+ 140t2 144(−1 + t)2t2 . Proposition 6.9. The graded Q-algebra of cusp forms with respect to Γ(6, 1) 6.4 Automorphic forms, Hecke operators and q-expansions 241 is generated by the eigenforms h4 = (Dt+6 ) 2￿ 3(t+6 − 1)t+6 (t+6 − 1) = 8√ 3 (Dt6)2 t6(t26 − 1) , h6 = 1 3 (Dt+6 ) 3￿ −t+6 t+6 (t+6 − 1)2 = 32i 3 (Dt6)3 t6(t26 − 1)2 , h12 = 1 3 (Dt+6 ) 6￿ −3t+6 (t+6 − 1)(t+6 )2(t+6 − 1)4 = 512 √−3 9 (Dt6)6(t26 + 1) t36(t 2 6 − 1)4 , which satisfy the relation 0 = h64 + 3h46 + h212. Proof. The expressions for these forms are obtained thanks to Proposition 6.5, which ensures as well that they are defined over Q. Moreover the relation between these forms follows immediately from the given expressions. In order to prove that they are indeed generators of the whole algebra, we will use Proposition 6.6. In this case, it is clear that h64, h46 are not proportional, that the ideal I = (x61+3x42+x123 ) is prime in Q[x1, x2, x3], since the generating polynomial is irreducible, and that the Krull dimension of the quotient is 2. Therefore, it is clear that Q[x1, x2, x3]/(x61 + 3x42 + x123 ) ∼= Q[h4, h6, h12] and it only remains to see that the dimension of the subspace of weigth k has the same dimension as Sk(Γ(6, 1))Q. This can be done using a computational algebra software, but in this case, since we only have one relation it is easily seen that they agree up to degree 12 and they increase in 2 when the weight increases in 12 (cf. Theorem 1.20 for the dimension of the space Sk(Γ(6, 1))Q). Now, h4 and h6 are obviously eigenforms, because the space of automorphic forms of weights 4 and 6 are both 1-dimensional. On the other hand, h12 is also an eigenform, since it lies in the 1-dimensional space S12(￿Γ(6, 1),ω3￿)−Q . Finally, in order to compute the expansions of these forms around a CM point P all we need is the expansion of t+6 around this point. In particular, if P is rational, it can be obtained using the information on Table 6.13. To show how the coefficients of these expansions look like we will give expansions around the following points inH: Q+6,1 = i, an elliptic point of order 2, cf. Table 6.22; Q+6,2 = −1+i 1+ √ 3 , an elliptic point of order 3, cf. Table 6.23, and Q+6,4 = 12 i ￿−√2 +√6￿, a CM point by the ring of integers of Q( √−6), cf. Table 6.24. 242 Computational applications: kroneckerian polynomials and expansions of automorphic functions and forms c −2 −1 0 1 2 3 4 5 6 7 8 9 h4 48 0 0 1 0 0 0 −30 0 0 0 −6120 0 h6 576i 0 0 0 1 0 0 0 84 0 0 0 −40824 h12 110592i 0 0 1 0 0 0 234 0 0 0 61776 0 Table 6.22: Expansions around Q+6,1 = i with κQ+6,1 = 1−i 4 π−4 c 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 h4 −4i √ 3 0 1 0 0 0 0 0 −5 0 0 0 0 0 −4980 0 0 h6 24 1 0 0 0 0 0 7 0 0 0 0 0 −84 0 0 0 h12 576i √ 3 1 0 0 0 0 0 −26 0 0 0 0 0 −18252 0 0 0 Table 6.23: Expansions around Q+6,2 with κQ+6,2 = − 1+i 3√18π−3 c 0 1 2 3 4 5 6 7 8 h4 96i √ 3 1 0 −10 0 −270 0 −13320 0 1371960 h6 −1152 √ 2 1 0 21 0 −252 0 40572 0 352296 h12 −15925248i √ 6 0 1 0 13 0 3432 0 −8424 0 Table 6.24: Expansions around Q+6,4 = 12 i ￿−√2 +√6￿ with κQ+6,4 = 1 12 √ 2 iπ−24 6.4 Automorphic forms, Hecke operators and q-expansions 243 Similarly, we can deal with the case of level N = 5. The Shimura curve of discriminant D = 6 and level N = 5 is a genus 1 curve defined over Q which has been studied in detail in Section 4.3.3, where a Hauptmodul for the curve X(6, 5)+ together with its automorphic derivative have been computed. Moreover, generating functions for all the fields Q(X(6, 5)W ) have also been computed. In particular, those corresponding to subgroups W ⊂ W6,5 of index 2 will be used to prove the following result. Proposition 6.10. The graded Q-algebra of cusp forms with respect to Γ(6, 5) is generated by the eigenforms f2 = Dt+6,5 2 ￿ 61− 64t+6,5 ￿ t+6,5 − 1 ￿ t+6,5 + 1 , f4,1 = (Dt+6,5) 2 4(t+6,5 − 1) ￿ t+6,5 + 1(−61 + 64t+6,5) , f4,2 = (Dt+6,5) 2 4(1 + t+6,5)(−61 + 64t+6,5) ￿ t+6,5 − 1 , f4,3 = (Dt+6,5) 2 4 ￿ 61− 64t+6,5 ￿ t+6,5 − 1 ￿ ￿ t+6,5 + 1 ￿ , which satisfy the relations 0 = f24,1 − f24,2 − 2f42 , 0 = 64f24,1 + f24,2 − 125f42 . The subalgebra of cusp forms for Γ(6, 1), S∗(Γ(6, 1))Q ⊂ S∗(Γ(6, 5))Q, is explicitly determined by the equalities h4 = 12(6f4,1 + f4,3), h6 = −72(7f4,1 − 3f4,3)f2, h12 = 1728(4250f 4 2 − 352f24,1 − 117f4,1f4,3)f4,2. Proof. The functions above can be obtained as in the case of level 1 by means of Proposition 6.5 and the relations can be easily checked using the expression of these functions. In order to prove that they are generators, we use again Proposition 6.6. In this case, the verifications are not so easy as in the previous case, but they can 244 Computational applications: kroneckerian polynomials and expansions of automorphic functions and forms be carried out by hand. However, they can also be checked immediately using a computational algebra software such as Magma or Macaulay2. Finally, to prove that all these forms are eigenforms we can use, as in the previous case, the Atkin-Lehner involutions. In the following table, for a given form, we will write +, resp.−, if the form is an eigenvector of eigenvalue 1, resp.−1: f22 f4,1 f4,2 f4,3 ω2 + − − − ω6 + + − + ω5 + + − − Therefore, there exists a unique (up to scalar factors) common basis of eigen- vectors for these involutions, which implies that they are indeed eigenforms. The identification of the generators of S∗(Γ(6, 1))Q as elements in S∗(Γ(6, 5))Q can be carried out easily comparing their expansions around a given CM point, for example, any of the ones which we are going to compute next. In order to complete our exposition we will give the first few terms of the expansions (following the same notations as in the previous cases) of these func- tions and forms around some special points, namely P4 = (2 + √ 3)i, an elliptic point of order 2, P4·52 = i, a CM point by Q(i) of level 5, P120 = −1+ √ 10i 1+2 √ 3 , a CM point by the ring of integers of Q( √−30), P40 = 3+ √ 30i 4 √ 3−3 , a CM point by the ring of integers of Q( √−10), P24 = √ 2i 1+ √ 3 , a CM point by the ring of integers of Q( √−6), and P19 √ 3−i√19 −5+√3 , a CM point by the ring of integers of Q( √−19). Note that in this case we can use Table 6.13 together with the relation between the functions t+6,5 and t + 6 given in Proposition 4.11 to obtain the first coefficients of the expansion of t+6,5 around any point of X(6, 5)+ related to a point in the table and then use the automorphic derivative to compute as many coefficients as desired. This also allows us to obtain the values of the local constants. 6.4 Automorphic forms, Hecke operators and q-expansions 245 c 0 1 2 3 4 5 6 7 8 f2 2 0 1 0 0 0 −474 0 0 0 f4,1 i/2 1 0 0 0 −1170 0 0 0 483480 f4,2 −i/2 1 0 0 0 366 0 0 0 2934936 f4,3 −4 1 0 0 0 330 0 0 0 4767480 Table 6.25: Expansions around P4 = (2 + √ 3)i with κP4 = 2−i10√6π−4 c 0 1 2 3 4 f2 3i √ 5 1 −9 45 29160 1212570 f4,1 27 √ 5 1 −25 −450 −8775 −4872150 f4,2 9i √ 5 1 45 2160 −93555 −1630530 f4,3 9 √ 5 7 450 8100 157950 −102145050 Table 6.26: Expansions around P4·52 = i with κP4·52 = i 30 4 √−180π−4 c 0 1 2 3 4 5 f2 −2 √ 2 1 0 43 0 29094 0 f4,1 −5 √ 5 1 0 110 0 −9630 0 f4,2 −i √ 3 1 0 −914 0 285282 0 f4,3 −80i √ 30 0 1 0 −675 0 −181980 Table 6.27: Expansions around P120 = −1+ √ 10i 1+2 √ 3 with κP120 = −√10+2√30−10i−2√−3 1320 π−120 246 Computational applications: kroneckerian polynomials and expansions of automorphic functions and forms c 0 1 2 3 4 f2 −i √ 2 1 0 83 0 −6666 f4,1 20 √ 5 0 1 0 75 0 f4,2 2i √ 2 1 0 −84 0 14502 f4,3 10 √ 5 1 0 −90 0 9570 Table 6.28: Expansions around P40 = 3+ √ 30i 4 √ 3−3 with κP40 = 12−10√3−3i√10−4i√30 1560 π−40 c 0 1 2 3 4 f2 − √ 2 1 0 13 0 −138 f4,1 −2 √ 2 1 0 20 0 5670 f4,2 −4i √ 3 0 1 0 −171 0 f4,3 2i √ 3 1 0 −230 0 19170 Table 6.29: Expansions around P24 = √ 2i 1+ √ 3 with κP24 = 3 √ 2−2√−2+2√3−2√−3 120 π−24 c 0 1 2 3 4 f2 −3i √ 19 1 27 −1044 63666 −37804968 f4,1 513/2 1 35 2790 −599940 4665600 f4,2 −171/2 1 −117 4158 −152604 −121803264 f4,3 171i √ 19 1 −90 7290 588060 43545600 Table 6.30: Expansions around P19 = √ 3−i√19 −5+√3 with κP19 = −3(7+19 √ 3)+i(9 √ 19−7√57) 37620 π−19 6.4 Automorphic forms, Hecke operators and q-expansions 247 Now we are going to deal with the cases N = 7, 11. Both of them are analogous to the cases above. Proposition 6.11. The graded Q-algebra of cusp forms with respect to Γ(6, 7) is generated by the forms f2 = Dt+6,7￿ 3t+6,7 ￿ 3t+6,7 − 3 ￿ −t+6,7 + 3281 , f4,1 = (Dt+6,7) 2￿ 3t+6,7(3t + 6,7 − 3)(−t+6,7 + 3281 ) , f4,2 = (Dt+6,7) 2 3t+6,7 ￿ 3t+6,7 − 3(−t+6,7 + 3281 ) , f4,3 = (Dt+6,7) 2 3t+6,7(3t + 6,7 − 3) ￿ −t+6,7 + 3281 , f6,1 = (Dt+6,7) 3 3t+6,7(3t + 6,7 − 3)(−t+6,7 + 3281 ) ￿ −t+6,7 + 3281 , f6,2 = (Dt+6,7) 3 3t+6,7(3t + 6,7 − 3)(−t+6,7 + 3281 ) ￿ 3t+6,7 − 3 , f6,3 = (Dt+6,7) 3 3t+6,7(3t + 6,7 − 3)(−t+6,7 + 3281 ) ￿ 3t+6,7 , f6,4 = (Dt+6,7) 3 (3t+6,7 − 3)(−t+6,7 + 3281 ) ￿ 3t+6,7 ￿ 3t+6,7 − 3 ￿ −t+6,7 + 3281 , which satisfy the relations 0 = −f4,1f4,2 + f2f6,1, 0 = −f4,1f4,3 + f2f6,2, 0 = −f4,2f4,3 + f2f6,3, 0 = −f24,1 + f2f6,4, 0 = 3f42 − f24,1 + f24,2, 0 = −32f42 + 27f24,1 + 81f24,3, 0 = f4,1f6,2 − f4,3f6,4, 0 = f4,3f6,1 − f4,2f6,2, 0 = f4,3f6,1 − f4,1f6,3, 0 = −f4,1f6,1 + f4,2f6,4, 0 = −f6,1f6,2 + f6,3f6,4, 248 Computational applications: kroneckerian polynomials and expansions of automorphic functions and forms 0 = −3f32 f4,3 + f4,1f6,2 − f4,2f6,3, 0 = −(32/27)f32 f4,2 + 3f4,3f6,3 + f4,2f6,4, 0 = 3f32 f4,1 + f4,2f6,1 − f4,1f6,4, 0 = −(32/27)f32 f4,1 + 3f4,3f6,2 + f4,1f6,4, 0 = 3f22 f4,1f4,3 + f6,1f6,3 − f6,2f6,4, 0 = −(32/27)f22 f4,1f4,2 + 3f6,2f6,3 + f6,1f6,4, 0 = 3f22 f 2 4,1 + f 2 6,1 − f26,4, 0 = (32/27)f62 − (32/81)f22 f24,1 + (1/3)f26,1 + f26,3, 0 = (49/81)f22 f 2 4,1 + (1/3)f 2 6,1 + f 2 6,2. A basis of eigenforms for S4(Γ(6, 7))Q is (f22 , f4,1, f4,2, f4,3) and one for S6(Γ(6, 7))Q is given by (f2f4,1, f2f4,2, f2f4,3, f6,1, f6,2, f6,3, −3f6,4 + 8f32 , 5f6,4 + 34f32 ). The subalgebra of cusp forms with respect to Γ(6, 1) inside S∗(Γ(6, 7))Q is explicitly determined by the equalities h4 = −20/3f4,2 + 18f4,3, h6 = −136/3f32 − 20/3f6,4 + 18f6,3, h12 = 1508f6,1f6,3 − 2793f6,1f6,4 − 9099f6,2f6,3 − 5096/3f6,2f6,3. Proof. In this case, everything can be done as in the two previous cases, except for finding eigenforms in the space S6(￿Γ(6, 7),ω2,ω7￿)−Q = ￿f32 , f6,4￿Q. In order to do so, we can use the expansions for t+6 and t + 6 ◦ ω for ω cor- responding to a set of representatives of O(D, 1)1\O(D, 1)5 and then use the relation between t+6 and t + 6,7 to obtain expansions for t + 6,7 and t + 6,7 ◦ ω. From these we can deduce expansions for f2 and f6,4 and T5f32 , T5f6,4. Comparing those, we find that −3f6,4+8f32 , 5f6,4+34f32 are eigenforms of eigenvalues −76 and 66, respectively. 6.4 Automorphic forms, Hecke operators and q-expansions 249 Similarly, from expansions for t+6,7 and t + 6,7 ◦ ω5 we could compute the poly- nomial given in Proposition 6.7, starting from a polynomial with indeterminate coefficients in the corresponding spaces of cusp forms of weights given in the proposition. Then, the action of T5 can be read from the coefficients. Proposition 6.12. The graded Q-algebra of cusp forms with respect to Γ(6, 11) is generated by the eigenforms f2,1 = Dt+6,11￿ −2t+6,11 + 2 ￿ −1331(t+6,11)2 − 118t+6,11 − 3 , f2,2 = Dt+6,11￿ −t+6,11 ￿ −1331(t+6,11)2 − 118t+6,11 − 3 , f2,3 = Dt+6,11￿ −t+6,11 ￿ −2t+6,11 + 2 ￿ −1331(t+6,11)2 − 118t+6,11 − 3 , f4 = (Dt+6,11) 2 −t+6,11(−2t+6,11 + 2) ￿ −1331(t+6,11)2 − 118t+6,11 − 3 , which satisfy the relations 0 = −2f22,1 + f22,2 − 2f22,3, 0 = f24 + 3f42,3 − 118f22,3f22,1 + 1331f42,1. The subalgebra of cusp forms with respect to Γ(6, 1) inside S∗(Γ(6, 11))Q is explicitly determined by the equalities h4 = 288f2,2f2,3 − 120f4, h6 =− 69408f22,2f2,3 − 3024f2,2f4 + 139392f32,3, h12 =− 7514726400f2,1f32,2f22,3 − 968827392f2,1f22,2f2,3f4 + 15736799232f2,1f2,2f 4 2,3 + 38541312f2,1f2,2f 2 4 + 2348476416f2,1f 3 2,3f4. 6.4.2 Cusp forms attached to Shimura curves of discrimi- nant 10 In this section we are going to provide generators for the graded algebras of cusp forms corresponding to the Shimura curves of discriminant 10 and levels 250 Computational applications: kroneckerian polynomials and expansions of automorphic functions and forms N = 1, 3, 7, 9. All the results we give are checked in the same way we dealt with the cases of discriminant 6. We start with the case of level 1. Proposition 6.13. The graded Q-algebra of cusp forms with respect to Γ(10, 1) is generated by the eigenforms f4 = (Dt+10) 2 (−2t+10)(2t+10 − 2)(t+10 − 27/2) , f6,1 = (Dt+10) 3 −2t+10(2t+10 − 2) ￿ −2t+10 ￿ 2t+10 − 2(t+10 − 27/2)2 , f6,2 = (Dt+10) 3 −2t+10(2t+10 − 2) ￿ 2t+10 − 2(t+10 − 27/2)2 , f6,3 = (Dt+10) 3 −2t+10(2t+10 − 2) ￿ −2t+10(t+10 − 27/2)2 , which satisfy the relations 0 = f26,3 + f 2 6,2 + 2f 2 6,1, 0 = f 3 4 + 27/2f 2 6,1 + 1/2f 2 6,2. In the case N = 3, we have the following result. Proposition 6.14. The graded Q-algebra of cusp forms with respect to Γ(10, 3) is generated by f2 = (Dt+10,3) ( ￿ −5t+10,3 ￿ 3t+10,3 − 3 ￿ 30t+10,3 − 75/16) , f4,1 = (Dt+10,3) 2 −5t+10,3 ￿ −5t+10,3 ￿ 3t+10,3 − 3(30t+10,3 − 75/16) , f4,2 = (Dt+10,3) 2 −5t+10,3 ￿ −5t+10,3(3t+10,3 − 3) ￿ 30t+10,3 − 75/16 , f4,3 = (Dt+10,3) 2 −5t+10,3 ￿ −5t+10,3(3t+10,3 − 3)(30t+10,3 − 75/16) , 6.4 Automorphic forms, Hecke operators and q-expansions 251 f6,1 = (Dt+10,3) 3 (−5t+10,3)2 ￿ −5t+10,3(3t+10,3 − 3) ￿ 3t+10,3 − 3 ￿ 30t+10,3 − 75 , f6,2 = (Dt+10,3) 3 (−5t+10,3)2 ￿ −5t+10,3(3t+10,3 − 3) ￿ 3t+10,3 − 3(30t+10,3 − 75) , f6,3 = (Dt+10,3) 3 (−5t+10,3)2 ￿ −5t+10,3(3t+10,3 − 3)(30t+10,3 − 75) , f6,4 = (Dt+10,3) 3 (−5t+10,3)2 ￿ −5t+10,3(3t+10,3 − 3)(30t+10,3 − 75) ￿ 30t+10,3 − 75 , which satisfy the relations 0 = f24,2 − f2f6,1, 0 = f4,1f4,2 − f2f6,3, 0 = f4,1f4,3 − f2f6,4, 0 = f4,2f4,3 − f2f6,2, 0 = 3f42 + 5f 2 4,1 + 15f 2 4,3, 0 = 96f42 + 16f 2 4,2 + 75f 2 4,3, 0 = f4,2f6,3 − f4,1f6,1, 0 = f4,2f6,4 − f4,1f6,2, 0 = −f4,3f6,3 + f4,2f6,4, 0 = f4,3f6,1 − f4,2f6,2, 0 = f6,4f6,1 − f6,3f6,2, 0 = 96f32 f4,2 + 16f4,2f6,1 + 75f4,3f6,2, 0 = 3f32 f4,2 + 5f4,1f6,3 + 15f4,3f6,2, 0 = 81f32 f4,3 − 25f4,1f6,4 + 16f4,2f6,2, 0 = 96f32 f6,3 + 16f6,3f6,1 + 75f6,4f6,2, 0 = 160f26,3 − 16f26,1 + 405f26,2, 0 = −25f26,3 + 81f32 f6,1 + 16f26,1, 0 = −25f6,3f6,4 + 81f32 f6,2 + 16f6,1f6,2, 0 = 96f32 f4,1 + 75f4,3f6,4 + 16f4,1f6,1, 0 = 209952f62 + 23600f 2 6,3 + 50625f 2 6,4 − 8192f26,1. 252 Computational applications: kroneckerian polynomials and expansions of automorphic functions and forms Next, we move to the case of level 7, for which 16 generators and 104 relations are needed. Proposition 6.15. The graded Q-algebra of cusp forms with respect to Γ(10, 7) is generated by f2 = Dt+10,7￿ −5t+10,7 ￿ 70(t+10,7 − 25/32) ￿ 7(t+10,7 − 1) , f4,1 = (Dt+10,7) 2 −5t+10,7(70(t+10,7 − 25/32)) ￿ 7(t+10,7 − 1)(t+10,7 − 20/27) , f4,2 = (Dt+10,7) 2 −5t+10,7(70(t+10,7 − 25/32))(7(t+10,7 − 1))(t+10,7 − 20/27) , f4,3 = (Dt+10,7) 2 −5t+10,7 ￿ 70(t+10,7 − 25/32) ￿ 7(t+10,7 − 1)(t+10,7 − 20/27) , f4,4 = (Dt+10,7) 2 −5t+10,7 ￿ 70(t+10,7 − 25/32)(7(t+10,7 − 1))(t+10,7 − 20/27) , f4,5 = (Dt+10,7) 2￿ −5t+10,7(70(t+10,7 − 25/32)) ￿ 7(t+10,7 − 1)(t+10,7 − 20/27) , f4,6 = (Dt+10,7) 2￿ −5t+10,7(70(t+10,7 − 25/32))(7(t+10,7 − 1))(t+10,7 − 20/27) , f4,7 = (Dt+10,7) 2￿ −5t+10,7 ￿ 70(t+10,7 − 25/32)(7(t+10,7 − 1))(t+10,7 − 20/27) , f6,1 = (Dt+10,7) 3 −5t+10,7(70(t+10,7 − 25/32))(7(t+10,7 − 1))(t+10,7 − 20/27)2 , f6,2 = (Dt+10,7) 3 −5t+10,7(70(t+10,7 − 25/32))(7(t+10,7 − 1)) ￿ 7(t+10,7 − 1)(t+10,7 − 20/27)2 , f6,3 = (Dt+10,7) 3 −5t+10,7(70(t+10,7 − 25/32)) ￿ 70(t+10,7 − 25/32)(7(t+10,7 − 1))(t+10,7 − 20/27)2 , f6,4 = (Dt+10,7) 3 −5t+10,7(70(t+10,7 − 25/32)) ￿ 70(t+10,7 − 25/32)(7(t+10,7 − 1)) ￿ 7(t+10,7 − 1)(t+10,7 − 20/27)2 , f6,5 = (Dt+10,7) 3 −5t+10,7 ￿ −5t+10,7(70(t+10,7 − 25/32))(7(t+10,7 − 1)) ￿ 7(t+10,7 − 1)(t+10,7 − 20/27)2 , f6,6 = (Dt+10,7) 3 −5t+10,7 ￿ −5t+10,7(70(t+10,7 − 25/32))(7(t+10,7 − 1))(t+10,7 − 20/27)2 , 6.4 Automorphic forms, Hecke operators and q-expansions 253 f6,7 = (Dt+10,7) 3 −5t+10,7 ￿ −5t+10,7(70(t+10,7 − 25/32)) ￿ 70(t+10,7 − 25/32)(7(t+10,7 − 1)) ￿ 7(t+10,7 − 1) · 1 (t+10,7 − 20/27)2 , f6,8 = (Dt+10,7) 3 −5t+10,7 ￿ −5t+10,7 ￿ 70(t+10,7 − 25/32)(7(t+10,7 − 1))(t+10,7 − 20/27)2 , which satisfy the relations 0 = f4,2f4,7 − f2f6,2, 0 = f4,1f4,7 − f2f6,1, 0 = f4,4f4,6 − f2f6,2, 0 = f4,3f4,6 − f2f6,1, 0 = f4,2f4,6 − f2f6,4, 0 = f4,1f4,6 − f2f6,3, 0 = f4,4f4,5 − f2f6,1, 0 = f4,2f4,5 − f2f6,3, 0 = f4,3f4,4 − f2f6,8, 0 = f4,2f4,4 − f2f6,5, 0 = f4,1f4,4 − f2f6,6, 0 = f4,2f4,3 − f2f6,6, 0 = f24,2 − f2f6,7, 0 = f4,7f6,6 − f4,6f6,8, 0 = f4,4f6,6 − f4,2f6,8, 0 = f4,3f6,6 − f4,1f6,8, 0 = f4,2f6,6 − f4,3f6,7, 0 = f4,6f6,5 − f4,7f6,7, 0 = f4,5f6,5 − f4,6f6,6, 0 = f4,3f6,5 − f4,2f6,8, 0 = f4,2f6,5 − f4,4f6,7, 0 = f4,1f6,5 − f4,3f6,7, 0 = f4,4f6,4 − f4,7f6,7, 0 = f4,3f6,4 − f4,6f6,6, 0 = f4,2f6,4 − f4,6f6,7, 0 = f4,1f6,4 − f4,5f6,7, 0 = f4,4f6,3 − f4,6f6,6, 0 = f4,3f6,3 − f4,5f6,6, 0 = f4,2f6,3 − f4,5f6,7, 0 = f4,4f6,2 − f4,7f6,5, 0 = f4,3f6,2 − f4,6f6,8, 0 = f4,2f6,2 − f4,7f6,7, 0 = f4,1f6,2 − f4,6f6,6, 0 = f4,4f6,1 − f4,6f6,8, 0 = f4,3f6,1 − f4,5f6,8, 0 = f4,2f6,1 − f4,6f6,6, 0 = f4,1f6,1 − f4,5f6,6, 0 = f6,5f6,6 − f6,7f6,8, 0 = f6,4f6,6 − f6,1f6,7, 0 = f6,2f6,6 − f6,4f6,8, 0 = f6,1f6,6 − f6,3f6,8, 0 = f6,4f6,5 − f6,2f6,7, 0 = f6,3f6,5 − f6,1f6,7, 0 = f6,1f6,5 − f6,4f6,8, 0 = 30240f22 f4,6 − 432f4,4f4,7 − 1225f2f6,4, 0 = 160f24,5 + 245f 2 4,6 − 16f24,7, 0 = 27f4,3f4,5 − 189f22 f4,7 + 49f2f6,2, 0 = 160f4,1f4,5 − 16f4,4f4,7 + 245f2f6,4, 0 = 30240f22 f4,5 − 432f4,3f4,7 − 1225f2f6,3, 0 = 16f24,4 + 224f 2 4,6 + 875f2f6,7, 0 = 135f22 f4,4 + 27f4,6f4,7 + 100f2f6,5, 0 = 80f24,3 − 7840f24,6 + 112f24,7 − 30625f2f6,7, 0 = 5f4,1f4,3 + 7f4,6f4,7 + 35f2f6,5, 0 = 135f22 f4,3 + 27f4,5f4,7 + 100f2f6,6, 0 = 875f4,1f4,2 + 224f4,5f4,6 + 16f2f6,8, 0 = 135f22 f4,2 + 27f 2 4,6 + 100f2f6,7, 0 = 5f24,1 + 7f 2 4,6 + 35f2f6,7, 0 = 4725f22 f4,1 + 49f4,5f4,6 − 64f2f6,8, 0 = 30240f22 f6,6 − 1225f4,3f6,7 − 432f4,4f6,8, 254 Computational applications: kroneckerian polynomials and expansions of automorphic functions and forms 0 = 432f4,4f6,5 − 30240f22 f6,7 + 1225f4,2f6,7, 0 = 189f22 f6,5 − 27f4,1f6,6 − 49f4,4f6,7, 0 = 7f4,7f6,4 + 5f4,1f6,6 + 35f4,4f6,7, 0 = 27f4,6f6,4 + 135f 2 2 f6,7 + 100f4,2f6,7, 0 = 224f4,5f6,4 + 875f4,1f6,7 + 16f4,2f6,8, 0 = 30240f22 f6,4 − 432f4,7f6,5 − 1225f4,6f6,7, 0 = 224f4,7f6,3 + 875f4,3f6,7 + 16f4,4f6,8, 0 = 224f4,6f6,3 + 875f4,1f6,7 + 16f4,2f6,8, 0 = 160f4,1f6,3 − 16f4,7f6,5 + 245f4,6f6,7, 0 = 30240f22 f6,3 − 1225f4,5f6,7 − 432f4,6f6,8, 0 = 7f4,6f6,2 + 5f4,1f6,6 + 35f4,4f6,7, 0 = 224f4,5f6,2 + 875f4,3f6,7 + 16f4,4f6,8, 0 = 189f22 f6,2 − 27f4,5f6,6 − 49f4,7f6,7, 0 = 27f4,7f6,1 + 135f 2 2 f6,8 + 100f4,2f6,8, 0 = 224f4,6f6,1 + 875f4,3f6,7 + 16f4,4f6,8, 0 = 224f4,5f6,1 + 875f4,1f6,6 + 16f4,3f6,8, 0 = 30240f22 f6,1 − 1225f4,6f6,6 − 432f4,7f6,8, 0 = 189f2f4,4f4,7 − 49f4,7f6,5 − 27f4,5f6,8, 0 = 1225f26,6 − 30240f2f4,1f6,8 + 432f26,8, 0 = 27f6,3f6,6 − 189f2f4,7f6,7 + 49f6,2f6,7, 0 = 432f26,5 − 30240f2f4,2f6,7 + 1225f26,7, 0 = 27f26,4 + 135f2f4,2f6,7 + 100f 2 6,7, 0 = 49f6,3f6,4 + 4725f2f4,1f6,7 − 64f6,7f6,8, 0 = 27f6,2f6,4 + 135f2f4,4f6,7 + 100f6,5f6,7, 0 = 224f6,1f6,4 + 875f6,6f6,7 + 16f6,5f6,8, 0 = 224f6,2f6,3 + 875f6,6f6,7 + 16f6,5f6,8, 0 = 27f6,1f6,2 + 135f2f4,2f6,8 + 100f6,7f6,8, 0 = 49f26,1 + 4725f2f4,1f6,8 − 64f26,8, 0 = 30240f2f4,5f6,7 − 1225f6,3f6,7 − 432f6,4f6,8, 0 = 30240f2f4,3f6,7 − 1225f6,6f6,7 − 432f6,5f6,8, 0 = 30240f2f4,6f6,6 − 1225f6,1f6,7 − 432f6,2f6,8, 0 = 432f6,2f6,5 − 30240f2f4,6f6,7 + 1225f6,4f6,7, 0 = 4082400f42 − 571725f24,6 + 11664f24,7 − 2240000f2f6,7, 0 = 6048f4,5f6,3 + 165375f 2 2 f6,7 − 42875f4,2f6,7 + 432f4,1f6,8, 0 = 3024f4,7f6,2 + 1058400f 2 2 f6,7 − 42875f4,2f6,7 + 2160f4,1f6,8, 0 = 42336f2f4,6f4,7 + 31465f4,1f6,6 + 54880f4,4f6,7 + 432f4,3f6,8, 0 = 1143072f2f 2 4,6 + 5946885f 2 2 f6,7 − 60025f4,2f6,7 + 11664f4,1f6,8, 0 = 193536f2f4,5f4,6 + 30625f4,1f6,7 + 13824f 2 2 f6,8 + 10800f4,2f6,8, 0 = 7001316000f32 f6,7−2098774125f2f4,2f6,7+73530625f26,7−352719360f2f4,1f6,8+5038848f26,8, 0 = 816480f2f4,5f6,6 − 231525f2f4,7f6,7 + 60025f6,2f6,7 − 11664f6,1f6,8, 0 = 816480f2f4,1f6,6 − 231525f2f4,4f6,7 + 60025f6,5f6,7 − 11664f6,6f6,8, 0 = 81648f2f4,7f6,5 − 1481760f2f4,6f6,7 + 60025f6,4f6,7 − 11664f6,3f6,8, 6.4 Automorphic forms, Hecke operators and q-expansions 255 0 = 7408800f26,3 + 202584375f2f4,2f6,7 − 52521875f26,7 + 13063680f2f4,1f6,8 − 186624f26,8, 0 = 6048f6,1f6,3 + 165375f2f4,4f6,7 − 42875f6,5f6,7 + 432f6,6f6,8, 0 = 740880f26,2 + 259308000f2f4,2f6,7 − 10504375f26,7 + 13063680f2f4,1f6,8 − 186624f26,8. Finally, we deal with the case N = 9. Note that a canonical model for the curve X(10, 9)+, which we did not give in Section 4.3.8, can be deduced from the equations given in the proposition. Proposition 6.16. The graded Q-algebra of cusp forms with respect to Γ(10, 9) is generated by the eigenforms f2,1 = Dt+10,9￿ t+10,9 + 3 ￿ t+10,9 − 6 ￿ t+10,9 + 2 ￿ t+10,9 − 1 , f2,2 = Dt+10,9￿ t+10,9 − 6 ￿ t+10,9 − 1 ￿ −3((t+10,9)2 − 3t+10,9 + 6 , f2,3 = Dt+10,9￿ t+10,9 + 2 ￿ t+10,9 − 1 ￿ −3((t+10,9)2 − 3t+10,9 + 6 , f2,4 = Dt+10,9￿ t+10,9 + 3 ￿ t+10,9 + 2 ￿ −3((t+10,9)2 − 3t+10,9 + 6 , f2,5 = Dt+10,9￿ t+10,9 + 3 ￿ t+10,9 − 6 ￿ −3((t+10,9)2 − 3t+10,9 + 6 , which satisfy the relations 0 = −3f22,3−5f22,2−f22,4+9f22,5, 0 = f22,2+2f22,4+f22,1, 0 = f2,2f2,4−f2,3f2,5. 6.4.3 Cusp forms attached to Shimura curves of discrimi- nant 22 Now we move on to the case of discriminant 22 and levels N = 1, 3, 5, 7. Ev- erything will follow the same pattern as for the other discriminants we have considered. We start with the case of level 1. 256 Computational applications: kroneckerian polynomials and expansions of automorphic functions and forms Proposition 6.17. The graded Q-algebra of cusp forms with respect to Γ(22, 1) is generated by the eigenforms f4,1 = (Dt+22) 2 −t+22(11t+22 − 11)(t+22 − 27/16) , f4,2 = (Dt+22) 2 −t+22(11t+22 − 11) ￿ 11t+22 − 11(t+22 − 27/16) , f4,3 = (Dt+22) 2￿ −t+22(11t+22 − 11) ￿ 11t+22 − 11(t+22 − 27/16) , f6,1 = (Dt+22) 3 −t+22 ￿ −t+22(11t+22 − 11)2(t+22 − 27/16)2 , f6,2 = (Dt+22) 3 −t+22 ￿ −t+22(11t+22 − 11)(t+22 − 27/16)2 , f6,3 = (Dt+22) 3 −t+22(11t+22 − 11) ￿ 11t+22 − 11(t+22 − 27/16)2 , f6,4 = (Dt+22) 3 −t+22(11t+22 − 11)2(t+22 − 27/16)2 , f6,5 = (Dt+22) 3 −t+22 ￿ −t+22(11t+22 − 11) ￿ 11t+22 − 11(t+22 − 27/16)2 , which satisfy the relations 0 = 11f24,3 + f 2 4,1 + 11f 2 4,2, 0 = −f4,2f6,2 + f4,1f6,5, 0 = −f4,2f6,5 + f4,1f6,1, 0 = f4,2f6,4 − f4,3f6,1, 0 = f4,1f6,4 − f4,3f6,5, 0 = −f4,2f6,3 + f4,1f6,4, 0 = f4,1f6,3 − f4,3f6,2, 0 = 11f4,3f6,4 + f4,1f6,5 + 11f4,2f6,1, 0 = 11f4,3f6,3 + f4,1f6,2 + 11f4,2f6,5, 0 = −f26,5 + f6,2f6,1, 0 = −f6,4f6,5 + f6,3f6,1, 0 = f6,4f6,2 − f6,3f6,5, 0 = −11f26,3 + 121f26,4 − f26,2 + 121f26,1, 0 = 11f6,3f6,4 + f6,2f6,5 + 11f6,5f6,1, 0 = 11f26,3 + f 2 6,2 + 11f 2 6,5, 0 = 176f34,1 − 121f26,3 − 27f26,2, 0 = 176f24,1f4,2 − 121f6,3f6,4 − 27f6,2f6,5, 0 = 1936f4,1f 2 4,2 + 297f 2 6,3 − 1331f26,4 + 27f26,2, 6.4 Automorphic forms, Hecke operators and q-expansions 257 0 = 176f4,3f4,1f4,2 − 16f6,4f6,2 + 121f6,4f6,1, 0 = 176f4,3f 2 4,1 − 16f6,3f6,2 + 121f6,3f6,1. Next, we give the result for level N = 3, where we already need 12 generators and 67 relations. Proposition 6.18. The graded Q-algebra of cusp forms with respect to Γ(22, 3) is generated by f2,1 = Dt+22,3￿ −2t+22,3 − 1 ￿ −11(t+22,3)2 + 2t+22 − 3 , f2,2 = Dt+22,3￿ −2t+22,3 − 1 ￿ −2t+22,3 + 3 ￿ −11(t+22,3)2 + 2t+22 − 3 , f2,3 = Dt+22,3￿ −2t+22,3 + 3 ￿ −11(t+22,3)2 + 2t+22 − 3 , f4,1 = (Dt+22,3) 2 (−2t+22,3 − 1) ￿ −2t+22,3 − 1 ￿ −2t+22,3 + 3 ￿ −11(t+22,3)2 + 2t+22 − 3 , f4,2 = (Dt+22,3) 2 (−2t+22,3 − 1) ￿ −2t+22,3 − 1(−2t+22,3 + 3) ￿ −11(t+22,3)2 + 2t+22 − 3 , f4,3 = (Dt+22,3) 2 (−2t+22,3 − 1)(−2t+22,3 + 3) ￿ −11(t+22,3)2 + 2t+22 − 3 , f4,4 = (Dt+22,3) 2 (−2t+22,3 − 1) ￿ −2t+22,3 − 1 ￿ −2t+22,3 + 3(−11(t+22,3)2 + 2t+22 − 3) , f4,5 = (Dt+22,3) 2 (−2t+22,3 − 1) ￿ −2t+22,3 − 1(−2t+22,3 + 3)(−11(t+22,3)2 + 2t+22 − 3) , f6,1 = (Dt+22,3) 3 (−2t+22,3 − 1)2 ￿ −2t+22,3 − 1 ￿ −2t+22,3 + 3(−11(t+22,3)2 + 2t+22 − 3) ￿ −11(t+22,3)2 + 2t+22 − 3 , f6,2 = (Dt+22,3) 3 (−2t+22,3 − 1)2 ￿ −2t+22,3 − 1(−2t+22,3 + 3)(−11(t+22,3)2 + 2t+22 − 3) ￿ −11(t+22,3)2 + 2t+22 − 3 , f6,3 = (Dt+22,3) 3 (−2t+22,3 − 1)2 ￿ −2t+22,3 − 1(−2t+22,3 + 3)(−11(t+22,3)2 + 2t+22 − 3) , f6,4 = (Dt+22,3) 3 (−2t+22,3 − 1)2 ￿ −2t+22,3 − 1(−2t+22,3 + 3) ￿ −2t+22,3 + 3(−11(t+22,3)2 + 2t+22 − 3) , which satisfy the relations 258 Computational applications: kroneckerian polynomials and expansions of automorphic functions and forms 0 = −f22,3 + f22,1 − 4f22,2, 0 = f32,2 − f2,3f4,5, 0 = f2,1f 2 2,2 − f2,3f4,4, 0 = f2,2f4,3 − f2,3f4,2, 0 = f2,1f4,3 − f2,3f4,1, 0 = −f2,2f4,4 + f2,1f4,5, 0 = −f2,2f4,1 + f2,1f4,2, 0 = f2,3f 2 2,2 − f2,1f4,4 + 4f2,2f4,5, 0 = f2,3f4,3 − f2,1f4,1 + 4f2,2f4,2, 0 = −f2,2f6,3 + f2,1f6,4, 0 = −f4,4f4,2 + f2,1f6,4, 0 = −f4,1f4,5 + f2,1f6,4, 0 = −f2,2f6,1 + f2,1f6,2, 0 = −f24,4 + f2,1f6,2, 0 = −f4,4f4,5 + f2,2f6,2, 0 = −f4,2f4,5 + f2,2f6,4, 0 = −f4,3f4,5 + f2,3f6,4, 0 = f4,3f4,4 − f2,3f6,3, 0 = −f4,1f4,4 + f2,1f6,3, 0 = −f2,1f2,2f4,2 + f2,3f6,3, 0 = −f2,1f2,2f4,5 + f2,3f6,2, 0 = −f22,2f4,2 + f2,3f6,4, 0 = −f22,1f4,5 + f2,3f6,1, 0 = f22,2f4,3 − f2,1f6,3 + 4f2,2f6,4, 0 = f24,3 − f24,1 + 4f24,2, 0 = f2,1f 3 2,2 − f2,1f6,1 + 4f2,2f6,2, 0 = 11f42,3 + 26f 2 2,3f 2 2,2 + 27f 4 2,2 + 4f 2 4,3, 0 = f2,1f 4 2,2 − f22,3f6,2, 0 = f32,2f4,3 − f22,3f6,4, 0 = f4,2f6,1 − f4,1f6,2, 0 = f4,2f6,1 − f4,4f6,3, 0 = f4,5f6,1 − f4,4f6,2, 0 = −f4,5f6,3 + f4,2f6,2, 0 = f4,2f6,2 − f4,4f6,4, 0 = f4,2f6,3 − f4,1f6,4, 0 = −f4,3f6,2 + f2,1f2,2f6,4, 0 = −f4,1f6,1 + 4f4,2f6,2 + f2,3f2,1f6,4, 0 = f22,3f 3 2,2 + 4f 5 2,2 − f22,3f6,1, 0 = −f4,4f6,3 + f2,3f2,2f6,4 + 4f4,5f6,4, 0 = −f4,3f6,1 + f22,3f6,4 + 4f22,2f6,4, 0 = −f6,3f6,2 + f6,1f6,4, 0 = −f6,1f6,3 + f2,3f4,4f6,4 + 4f6,2f6,4, 0 = −f26,1 + f2,3f4,4f6,2 + 4f26,2, 0 = −286f42,3 − 379f22,3f22,2 − 131f24,3 + 27f24,1 + 729f24,5, 0 = 44f22,3f2,1f2,2 + 77f2,1f 3 2,2 + 16f4,1f4,2 + 27f2,1f6,1, 0 = −847f42,3 − 814f22,3f22,2 − 416f24,3 + 108f24,1 + 729f2,1f6,2, 0 = 44f32,3f2,2 + 77f2,3f 3 2,2 + 16f4,3f4,2 + 27f2,3f6,1, 0 = 11f32,3f2,1 + 26f2,3f2,1f 2 2,2 + 4f4,3f4,1 + 27f2,3f6,2, 0 = −f2,3f2,1f32,2 + f2,3f2,1f6,1 − 4f4,4f6,1 + 16f4,5f6,2, 6.4 Automorphic forms, Hecke operators and q-expansions 259 0 = 99f2,3f2,1f 3 2,2 + 77f2,3f2,1f6,1 + 108f4,4f6,1 + 64f4,2f6,3, 0 = −11f42,3f2,1 − 26f22,3f2,1f22,2 − 27f2,1f42,2 − 4f2,1f24,1 + 16f4,3f6,3, 0 = −847f32,3f22,2 − 814f2,3f42,2 + 729f4,5f6,1 − 308f4,1f6,3 + 1664f4,2f6,4, 0 = −99f2,3f24,5 + 44f2,3f2,1f6,2 + 27f4,4f6,2 + 16f4,2f6,4, 0 = 99f2,2f 2 4,5 + 11f 2 2,3f6,1 − 18f2,1f2,2f6,2 + 4f4,3f6,4, 0 = 27f26,1 + 44f 2 2,3f2,1f6,2 + 77f2,3f4,4f6,2 + 16f2,2f4,3f6,4 + 64f 2 6,4, 0 = 44f22,3f2,2f6,2 + 77f2,3f4,5f6,2 + 27f6,1f6,2 + 16f6,3f6,4, 0 = 27f26,1 + 16f 2 6,3 + 44f 2 2,3f2,1f6,2 + 77f2,3f4,4f6,2. The result for N = 5 is very similar to the previous one, as can be seen in the following proposition. Proposition 6.19. The graded Q-algebra of cusp forms with respect to Γ(22, 5) is generated by f2,1 = Dt+22,5￿ 1375(t+22,5) 3 − 3650(t+22,5)2 + 3295t+22,5 − 1024 , f2,2 = Dt+22,5￿ −5t+22,5 ￿ −5t+22,54 ￿ 1375(t+22,5) 3 − 3650(t+22,5)2 + 3295t+22,5 − 1024 , f2,3 = Dt+22,5￿ −5t+22,5 ￿ 1375(t+22,5) 3 − 3650(t+22,5)2 + 3295t+22,5 − 1024 , f2,4 = Dt+22,5￿ −5t+22,54 ￿ 1375(t+22,5) 3 − 3650(t+22,5)2 + 3295t+22,5 − 1024 , f2,5 = ￿ −5t+22,5(Dt+22,5)￿ −5t+22,54 ￿ 1375(t+22,5) 3 − 3650(t+22,5)2 + 3295t+22,5 − 1024 , f4,1 = (Dt+22,5) 2 −5t+22,5 ￿ −5t+22,5 ￿ −5t+22,54(1375(t+22,5)3 − 3650(t+22,5)2 + 3295t+22,5 − 1024) , 260 Computational applications: kroneckerian polynomials and expansions of automorphic functions and forms f4,2 = (Dt+22,5) 2 −5t+22,5 ￿ −5t+22,5(1375(t+22,5)3 − 3650(t+22,5)2 + 3295t+22,5 − 1024) , f4,3 = (Dt+22,5) 2￿ −5t+22,5(−5t+22,54) ￿ 1375(t+22,5) 3 − 3650(t+22,5)2 + 3295t+22,5 − 1024 , f4,4 = (Dt+22,5) 2 −5t+22,5 ￿ −5t+22,5 ￿ 1375(t+22,5) 3 − 3650(t+22,5)2 + 3295t+22,5 − 1024 , f4,5 = (Dt+22,5) 2 −5t+22,5 ￿ −5t+22,54 ￿ 1375(t+22,5) 3 − 3650(t+22,5)2 + 3295t+22,5 − 1024 , f4,6 = (Dt+22,5) 2 −5t+22,5(−5t+22,54) ￿ 1375(t+22,5) 3 − 3650(t+22,5)2 + 3295t+22,5 − 1024 , f4,7 = (Dt+22,5) 2 −5t+22,5 ￿ −5t+22,5 ￿ −5t+22,54 ￿ 1375(t+22,5) 3 − 3650(t+22,5)2 + 3295t+22,5 − 1024 , which satisfy the relations 0 = −f2,4f2,3 + f2,1f2,2, 0 = f2,1f2,4 − f2,3f2,5, 0 = −f22,4 + f2,2f2,5, 0 = f22,1 − 4f22,3 + 16f22,2 − f22,5, 0 = −f2,1f2,3 + 4f2,4f2,2 + f2,4f2,5, 0 = −f22,1 + 4f22,4 + f22,5, 0 = −f2,1f4,3 + f4,5f2,5, 0 = −f2,4f4,3 + f4,6f2,5, 0 = −f2,1f4,5 + f4,4f2,5, 0 = −f2,1f2,3f2,2 + f4,2f2,5, 0 = −f2,1f4,6 + f4,7f2,5, 0 = −f2,4f4,5 + f4,7f2,5, 0 = −f2,3f4,3 + f4,7f2,5, 0 = −f2,1f22,2 + f4,1f2,5, 0 = −f2,2f4,3 + f2,4f4,6, 0 = −f2,4f4,4 + f2,1f4,7, 0 = −f2,3f4,5 + f2,1f4,7, 0 = −f2,4f4,2 + f2,1f4,1, 0 = −f2,3f4,6 + f2,4f4,7, 0 = −f2,2f4,5 + f2,4f4,7, 0 = −f2,3f22,2 + f2,4f4,1, 0 = −f2,2f4,4 + f2,3f4,7, 0 = −f2,2f4,2 + f2,3f4,1, 0 = −f2,3f4,2 + 4f2,2f4,1 + f4,1f2,5, 0 = −f2,3f4,4 + 4f2,2f4,7 + f4,7f2,5, 0 = 4f32,2 − f2,1f4,1 + f22,2f2,5, 0 = 4f2,2f4,6 − f2,1f4,7 + f4,6f2,5, 0 = −f2,1f4,2 + 4f2,4f4,1 + f2,3f2,2f2,5, 0 = −f2,1f4,4 + 4f2,4f4,7 + f4,5f2,5, 0 = 4f2,4f4,6 + f4,3f2,5 − f4,4f2,5, 0 = −f24,5 + f4,3f4,4, 0 = −f4,3f4,2 + f2,1f2,2f4,7, 0 = −f4,5f4,6 + f4,3f4,7, 6.4 Automorphic forms, Hecke operators and q-expansions 261 0 = f2,4f2,2f4,7 − f4,3f4,1, 0 = f4,6f4,4 − f4,5f4,7, 0 = f2,3f2,2f4,7 − f4,5f4,1, 0 = −f4,6f4,2 + f2,3f2,2f4,7, 0 = f22,2f4,7 − f4,6f4,1, 0 = −f4,2f4,7 + f4,4f4,1, 0 = −f4,4f4,2 + f2,1f2,2f4,7 + 4f4,7f4,1, 0 = f4,3f4,4 − f24,4 + 4f24,7, 0 = −f4,5f4,2 + 4f22,2f4,7 + f2,2f4,7f2,5, 0 = f4,3f4,5 − f4,5f4,4 + 4f4,6f4,7, 0 = f24,3 + 4f 2 4,6 − f4,3f4,4, 0 = 825f42,1 − 328f4,3f4,4 + 403f24,4 + 284777f2,1f2,3f4,2 + 1650688f24,1 + 25382f22,1f2,2f2,5, 0 = 659f2,1f2,2f4,2 + f4,4f4,7 + 1024f4,2f4,1 + 11f 2 2,1f2,3f2,5 + 146f2,1f4,2f2,5, 0 = −1991f42,1 − 584f4,3f4,4 + 403f24,4 + 116073f2,1f2,3f4,2 + 412672f24,2 − 730f22,1f2,2f2,5, 0 = −25382f32,1f2,2 − 256f4,3f4,5 + 5f4,5f4,4 + 20480f22,2f4,1 − 2761f32,1f2,5 − 60961f2,3f4,2f2,5, 0 = 11f32,1f2,3 + f4,6f4,4 + 102f 2 2,1f4,2 + 1024f 2 2,2f4,2 + 251f2,2f4,2f2,5, 0 = 146f32,1f2,2 + f4,5f4,4 + 1024f2,3f2,2f4,2 + 11f 3 2,1f2,5 + 659f2,3f4,2f2,5, 0 = −1727f42,1 + 32240f42,2 − 403f24,3 + 246f4,3f4,4 + 60961f2,1f2,3f4,2 + 1718f22,1f2,2f2,5, 0 = 251f2,1f2,2f4,2 + f4,3f4,7 + 20f2,4f2,2f4,1 + 11f 2 2,1f2,3f2,5 + 102f2,1f4,2f2,5, 0 = −11f42,1 − f4,3f4,4 − 659f2,1f2,3f4,2 + 1612f2,4f2,2f4,2 − 102f22,1f2,2f2,5, 0 = 11f32,1f2,3 + 20f2,4f 3 2,2 + f4,3f4,6 + 58f 2 2,1f4,2 + 19f2,2f4,2f2,5. Finally, we deal with the case of level 7, where 21 generators and 287 relations are needed. Proposition 6.20. The graded Q-algebra of cusp forms with respect to Γ(22, 7) is generated by f2,1 = Dt+22,7￿ 45(t+22,7) 2 − 48t+22,7 + 4 ￿ −3(48(t+22,7)2 − 56t+22,7 + 9) , f2,2 = Dt+22,7￿ 36(t+22,7) 2 − 36t+22,7 + 1 ￿ −3(48(t+22,7)2 − 56t+22,7 + 9) , f2,3 = Dt+22,7￿ 36(t+22,7) 2 − 36t+22,7 + 1 ￿ 45(t+22,7) 2 − 48t+22,7 + 4 , f2,4 = t+22,7(Dt + 22,7)￿ 36(t+22,7) 2 − 36t+22,7 + 1 ￿ 45(t+22,7) 2 − 48t+22,7 + 4 ￿ −3(48(t+22,7)2 − 56t+22,7 + 9) , 262 Computational applications: kroneckerian polynomials and expansions of automorphic functions and forms f2,5 = Dt+22,7￿ 36(t+22,7) 2 − 36t+22,7 + 1 ￿ 45(t+22,7) 2 − 48t+22,7 + 4 ￿ −3(48(t+22,7)2 − 56t+22,7 + 9) , f4,1 = (Dt+22,7) 2 t+22,7 ￿ 36(t+22,7) 2 − 36t+22,7 + 1 ￿ 45(t+22,7) 2 − 48t+22,7 + 4 ￿ −3(48(t+22,7)2 − 56t+22,7 + 9) , f4,2 = (Dt+22,7) 2 t+22,7 ￿ 36(t+22,7) 2 − 36t+22,7 + 1 ￿ 45(t+22,7) 2 − 48t+22,7 + 4(−3(48(t+22,7)2 − 56t+22,7 + 9)) , f4,3 = (Dt+22,7) 2 t+22,7 ￿ 36(t+22,7) 2 − 36t+22,7 + 1(45(t+22,7)2 − 48t+22,7 + 4) ￿ −3(48(t+22,7)2 − 56t+22,7 + 9) , f4,4 = (Dt+22,7) 2 t+22,7(36(t + 22,7) 2 − 36t+22,7 + 1) ￿ 45(t+22,7) 2 − 48t+22,7 + 4 ￿ −3(48(t+22,7)2 − 56t+22,7 + 9) , f4,5 = (Dt+22,7) 2 t+22,7(45(t + 22,7) 2 − 48t+22,7 + 4) ￿ −3(48(t+22,7)2 − 56t+22,7 + 9) , f4,6 = (Dt+22,7) 2 t+22,7 ￿ 36(t+22,7) 2 − 36t+22,7 + 1(−3(48(t+22,7)2 − 56t+22,7 + 9)) , f4,7 = (Dt+22,7) 2 t+22,7 ￿ 45(t+22,7) 2 − 48t+22,7 + 4(−3(48(t+22,7)2 − 56t+22,7 + 9)) , f4,8 = (Dt+22,7) 2 t+22,7(45(t + 22,7) 2 − 48t+22,7 + 4)(−3(48(t+22,7)2 − 56t+22,7 + 9)) , f6,1 = (Dt+22,7) 3 (t+22,7) 2(36(t+22,7) 2 − 36t+22,7 + 1) ￿￿ 45(t+22,7) 2 − 48t+22,7 + 4 ￿3 · 1￿￿ −3(48(t+22,7)2 − 56t+22,7 + 9) ￿3 , f6,2 = (Dt+22,7) 3 (t+22,7) 2 ￿￿ 36(t+22,7) 2 − 36t+22,7 + 1 ￿3 (45(t+22,7) 2 − 48t+22,7 + 4) · 1￿￿ −3(48(t+22,7)2 − 56t+22,7 + 9) ￿3 , f6,3 = (Dt+22,7) 3 (t+22,7) 2 ￿￿ 36(t+22,7) 2 − 36t+22,7 + 1 ￿3 ￿￿ 45(t+22,7) 2 − 48t+22,7 + 4 ￿3 · 1 (−3(48(t+22,7)2 − 56t+22,7 + 9) , 6.4 Automorphic forms, Hecke operators and q-expansions 263 f6,4 = (Dt+22,7) 3 (t+22,7) 2 ￿￿ 36(t+22,7) 2 − 36t+22,7 + 1 ￿3 ￿￿ 45(t+22,7) 2 − 48t+22,7 + 4 ￿3 · 1￿￿ −3(48(t+22,7)2 − 56t+22,7 + 9) ￿3 , f6,5 = (Dt+22,7) 3 (t+22,7) 2(36(t+22,7) 2 − 36t+22,7 + 1)(45(t+22,7)2 − 48t+22,7 + 4)(−3(48(t+22,7)2 − 56t+22,7 + 9) , f6,6 = (Dt+22,7) 3 (t+22,7) 2(36(t+22,7) 2 − 36t+22,7 + 1)(45(t+22,7)2 − 48t+22,7 + 4) ￿￿ −3(48(t+22,7)2 − 56t+22,7 + 9) ￿3 , f6,7 = (Dt+22,7) 3 (t+22,7) 2(36(t+22,7) 2 − 36t+22,7 + 1) ￿￿ 45(t+22,7) 2 − 48t+22,7 + 4 ￿3 (−3(48(t+22,7)2 − 56t+22,7 + 9) , f6,8 = (Dt+22,7) 3 (t+22,7) 2 ￿￿ 36(t+22,7) 2 − 36t+22,7 + 1 ￿3 (45(t+22,7) 2 − 48t+22,7 + 4)(−3(48(t+22,7)2 − 56t+22,7 + 9) , which satisfy the relations 0 = f2,1f4,6 − f4,7f2,2, 0 = f2,1f4,1 − f4,5f2,2, 0 = f2,1f4,1 − f4,7f2,3, 0 = f2,1f4,2 − f4,8f2,2, 0 = f2,1f4,2 − f4,7f2,5, 0 = f2,1f4,3 − f4,8f2,3, 0 = f2,1f4,3 − f4,5f2,5, 0 = f2,2f4,1 − f4,6f2,3, 0 = f2,2f4,2 − f4,6f2,5, 0 = f2,1f4,4 − f2,2f4,3, 0 = f2,1f4,4 − f2,3f4,2, 0 = f2,1f4,4 − f2,5f4,1, 0 = f4,7f4,1 − f4,5f4,6, 0 = f4,7f4,2 − f4,8f4,6, 0 = f4,8f4,1 − f4,5f4,2, 0 = f4,8f4,1 − f4,7f4,3, 0 = f4,1f4,2 − f4,6f4,3, 0 = f4,1f4,2 − f4,7f4,4, 0 = f4,1f4,3 − f4,5f4,4, 0 = f2,1f6,5 − f4,2f4,3, 0 = f2,1f6,5 − f4,8f4,4, 0 = f2,2f6,5 − f4,2f4,4, 0 = f2,3f6,5 − f4,3f4,4, 0 = f2,1f6,8 − f2,2f6,7, 0 = f2,1f6,8 − f2,3f6,6, 0 = f2,1f6,8 − f2,5f6,5, 0 = f2,1f6,2 − f2,2f6,1, 0 = f2,1f6,2 − f2,5f6,6, 0 = f2,1f6,3 − f2,3f6,1, 0 = f2,1f6,3 − f2,5f6,7, 0 = f2,1f6,4 − f2,5f6,1, 0 = f2,4f6,8 − f22,5f4,4, 0 = f2,2f6,3 − f2,3f6,2, 0 = f2,2f6,3 − f2,5f6,8, 0 = f2,2f6,4 − f2,5f6,2, 0 = f2,3f6,4 − f2,5f6,3, 0 = f4,5f6,6 − f4,8f6,5, 0 = f4,5f6,6 − f4,7f6,7, 0 = f4,8f6,6 − f4,7f6,1, 0 = f4,5f6,1 − f4,8f6,7, 0 = f4,1f6,6 − f4,2f6,5, 0 = f4,1f6,6 − f4,6f6,7, 0 = f4,1f6,6 − f4,7f6,8, 0 = f4,2f6,6 − f4,6f6,1, 0 = f4,2f6,6 − f4,7f6,2, 0 = f4,1f6,7 − f4,3f6,5, 0 = f4,1f6,7 − f4,5f6,8, 0 = f4,1f6,1 − f4,2f6,7, 0 = f4,1f6,1 − f4,3f6,6, 0 = f4,1f6,1 − f4,5f6,2, 0 = f4,1f6,1 − f4,8f6,8, 0 = f4,1f6,1 − f4,7f6,3, 0 = f4,2f6,1 − f4,8f6,2, 264 Computational applications: kroneckerian polynomials and expansions of automorphic functions and forms 0 = f4,2f6,1 − f4,7f6,4, 0 = f4,3f6,7 − f4,5f6,3, 0 = f4,3f6,1 − f4,5f6,4, 0 = f4,3f6,1 − f4,8f6,3, 0 = f4,1f6,8 − f4,4f6,5, 0 = f4,1f6,2 − f4,2f6,8, 0 = f4,1f6,2 − f4,4f6,6, 0 = f4,1f6,2 − f4,6f6,3, 0 = f4,2f6,2 − f4,6f6,4, 0 = f4,1f6,3 − f4,3f6,8, 0 = f4,1f6,3 − f4,4f6,7, 0 = f4,1f6,4 − f4,2f6,3, 0 = f4,1f6,4 − f4,3f6,2, 0 = f4,1f6,4 − f4,4f6,1, 0 = f6,5f6,1 − f6,6f6,7, 0 = f6,5f6,2 − f6,6f6,8, 0 = f6,5f6,3 − f6,7f6,8, 0 = f6,1f6,8 − f6,7f6,2, 0 = f6,1f6,8 − f6,6f6,3, 0 = f6,1f6,8 − f6,5f6,4, 0 = f6,1f6,2 − f6,6f6,4, 0 = f6,1f6,3 − f6,7f6,4, 0 = f6,2f6,3 − f6,8f6,4, 0 = f2,1f2,2f2,3 − f4,1f2,4, 0 = f2,1f2,2f2,5 − f2,4f4,2, 0 = f2,1f2,3f2,5 − f2,4f4,3, 0 = f2,2f2,3f2,5 − f2,4f4,4, 0 = f2,1f2,5f4,4 − f2,4f6,5, 0 = −6f22,1 + 8f22,2 + f22,3 + f22,5, 0 = −71f22,1 + 92f22,2 + 11f22,3 + 12f2,5f2,4, 0 = −52f22,1 + 67f22,2 + 8f22,3 + 9f22,4, 0 = −6f24,7 + 8f24,6 + f24,1 + f24,2, 0 = −71f24,7+12f2,2f2,5f4,7+92f24,6+11f24,1, 0 = −52f24,7 + 9f2,2f2,4f4,7 + 67f24,6 + 8f24,1, 0 = −6f2,1f4,7+8f2,2f4,6+f2,3f4,1+f2,5f4,2, 0 = 71f32,5 − 89460f22,4f2,5 + 92016f32,4 − 108f2,1f4,6 − 16f4,5f2,3 − 2567f4,8f2,4, 0 = 23004f32,4 − 23004f2,5f22,4 + 639f22,5f2,4 − 624f4,8f2,4 − 27f2,1f4,6 − 4f4,5f2,3, 0 = 373f2,3f 2 2,5 − 396f2,3f22,4 + 11f4,5f2,4 + 16f2,1f4,3 − 4f2,2f4,4, 0 = −3456f2,3f22,4 + 96f4,5f2,4 + 3357f2,3f2,5f2,4 + 4f2,1f4,3 − f2,2f4,4, 0 = 781f2,2f 2 2,5 − 828f2,2f22,4 + 23f4,7f2,4 + 27f2,1f4,2 + f2,3f4,4, 0 = −28944f2,2f22,4 + 804f4,7f2,4 + 28116f2,2f2,5f2,4 + 27f2,1f4,2 + f2,3f4,4, 0 = −36f2,2f2,3f2,5 + f4,4f2,5 − f2,1f4,1 + 36f2,2f2,3f2,4, 0 = 12f2,1f 2 2,5 − 71f2,1f4,8 + 92f2,2f4,2 + 11f2,3f4,3, 0 = −52f2,1f4,8 + 9f2,1f2,5f2,4 + 67f2,2f4,2 + 8f2,3f4,3, 0 = 135f2,1f 2 2,4 − 3f4,6f2,4 − 761f2,1f4,8 + 980f2,2f4,2 + 117f2,3f4,3, 0 = 4f2,1f4,5 − 3f2,2f4,1 − 9f2,1f2,3f2,4 + 8f2,5f4,3, 0 = 4f2,1f4,5 + 96f2,1f2,3f2,5 − f2,2f4,1 − 99f2,1f2,3f2,4, 0 = −71f2,1f4,7 + 92f2,2f4,6 + 12f2,1f2,2f2,5 + 11f2,3f4,1, 0 = −52f2,1f4,7 + 67f2,2f4,6 + 8f2,3f4,1 + 9f2,1f2,2f2,4, 6.4 Automorphic forms, Hecke operators and q-expansions 265 0 = 16f2,1f4,6 + 5f4,5f2,3 + 71f4,8f2,5 − 72f4,8f2,4, 0 = −8f4,5f4,8 − 207f2,3f2,4f4,8 − 5f4,1f4,2 − 384f2,2f2,5f4,4 + 576f2,2f2,4f4,4 + 28f2,3f6,4, 0 = −12f4,5f4,8 − 1623f2,3f2,4f4,8 + 3f4,1f4,2 − 128f2,2f2,5f4,4 + 1536f2,2f2,4f4,4 + 224f2,4f6,3, 0 = 94016f24,7−575f24,5−7448f24,8−123648f24,6−14676f24,1−84f24,4+3024f2,2f2,3f4,4+147f2,5f6,4, 0 = 441f42,5+267024f 2 4,7−1650f24,5−21266f24,8−351232f24,6−41684f24,1−245f24,4+8568f2,2f2,3f4,4, 0 = 12328f24,7+50f 2 4,5+1127f 2 4,8−15456f24,6−1788f24,1−84f24,4+1008f2,2f2,3f4,4+147f2,3f6,3, 0 = −448f4,7f4,8 + 576f4,6f4,2 + 69f4,1f4,3 + 12f2,3f2,5f4,4 − 36f2,3f2,4f4,4 + 7f2,2f6,4, 0 = −1309f4,7f4,8 + 1684f4,6f4,2 + 201f4,1f4,3 + 3f2,3f2,5f4,4 − 72f2,3f2,4f4,4 + 21f2,4f6,2, 0 = 4f4,7f4,5 − 3f4,6f4,1 − 9f4,7f2,3f2,4 + 288f2,5f2,4f4,4 − 288f2,4f6,8 + 8f2,2f6,3, 0 = −f4,1f4,3 − 36f2,3f2,5f4,4 + 36f2,3f2,4f4,4 + f2,3f6,8, 0 = −448f24,7 + 576f24,6 + 68f24,1 + f24,4 − 24f2,2f2,3f4,4 + 7f2,2f6,2, 0 = −f4,1f4,2 − 36f2,2f2,5f4,4 + 36f2,2f2,4f4,4 + f2,2f6,8, 0 = 192f2,1f4,8f2,3 − 384f4,6f2,5f2,3 + 576f4,6f2,4f2,3 − 7f4,7f4,1 − 405f2,1f4,5f2,4 + 28f2,1f6,3, 0 = 9f2,1f4,8f2,3 − 4f4,6f2,5f2,3 + 48f4,6f2,4f2,3 − 60f2,1f4,5f2,4 + 7f2,4f6,7, 0 = 29547f2,1f4,8f2,4+52416f4,6f2,5f2,4+729f4,7f4,2+12328f4,5f4,3−9152f4,1f4,4+24388f2,3f6,7, 0 = −199f4,7f4,6−4992f2,2f2,5f4,6+7488f2,2f2,4f4,6−16f4,5f4,1−2709f2,1f4,7f2,4+364f2,1f6,2, 0 = −81f4,7f4,6−832f2,2f2,5f4,6+9984f2,2f2,4f4,6−12f4,5f4,1−10563f2,1f4,7f2,4+1456f2,4f6,6, 0 = −60f2,1f4,7f2,3 + 48f2,2f4,6f2,3 + 9f4,8f4,1 − 4f4,6f4,4 + 7f2,1f6,8, 0 = −165f2,1f4,7f2,3 + 132f2,2f4,6f2,3 + 16f4,8f4,1 − 4f4,6f4,4 + 21f2,1f2,5f4,4, 0 = −72f4,6f2,5f2,4 + 14f4,7f4,2 + 3f4,1f4,4 + 67f2,2f6,6, 0 = 804f4,6f 2 2,5 − 828f4,6f2,4f2,5 + 27f4,7f4,2 + f4,1f4,4, 0 = −36f4,6f2,3f2,5 − f4,7f4,1 + 36f4,6f2,3f2,4 + f2,2f6,5, 0 = 1480f24,7 − 25f24,5 − 301f24,8 − 2016f24,6 − 240f24,1 + 42f2,1f6,1, 0 = 17288f24,7 − 275f24,5 − 3521f24,8 − 23520f24,6 + 504f4,8f22,5 − 2808f24,1, 0 = 6298f24,7 − 100f24,5 − 1288f24,8 − 8568f24,6 − 1023f24,1 + 189f4,8f2,5f2,4, 0 = 4f4,5f4,8 − 9f2,3f2,4f4,8 − 3f4,1f4,2 + 8f2,1f6,7, 0 = 4f4,5f4,8 + 96f2,3f2,5f4,8 − 99f2,3f2,4f4,8 − f4,1f4,2, 0 = 536f24,7 + 25f 2 4,5 + 49f 2 4,8 − 672f24,6 − 96f24,1 + 42f24,3, 0 = −6f4,7f4,8 + 8f4,6f4,2 + f4,1f4,3 + f2,1f6,6, 266 Computational applications: kroneckerian polynomials and expansions of automorphic functions and forms 0 = 12f4,7f 2 2,5 − 71f4,7f4,8 + 92f4,6f4,2 + 11f4,1f4,3, 0 = −52f4,7f4,8 + 9f4,7f2,5f2,4 + 67f4,6f4,2 + 8f4,1f4,3, 0 = 4f4,7f4,5 − 3f4,6f4,1 − 9f4,7f2,3f2,4 + 8f2,1f6,5, 0 = 4f4,7f4,5 + 96f4,7f2,3f2,5 − f4,6f4,1 − 99f4,7f2,3f2,4, 0 = 624f2,1f4,8f2,5 − 639f2,1f4,8f2,4 + 27f4,7f4,2 + 4f4,5f4,3, 0 = −48f2,1f4,8f2,3 − f4,7f4,1 + 45f2,1f4,5f2,4 + 4f4,8f4,3, 0 = 7f4,7f4,6 + 2f4,5f4,1 − 27f2,1f4,7f2,4 + 26f4,8f4,2, 0 = 27f4,7f4,6 + 624f2,1f4,7f2,5 + 4f4,5f4,1 − 639f2,1f4,7f2,4, 0 = 5184f6,8f 2 2,4 − 135f2,1f6,5f2,4 − 5040f2,5f6,8f2,4 + 3f4,6f6,5 − 16f4,1f6,1 + 4f4,4f6,4, 0 = 48f6,8f 2 2,5 − 1728f2,4f6,8f2,5 + f4,6f6,5 − 45f2,1f2,4f6,5 − 4f4,1f6,1 + 1728f22,4f6,8, 0 = −f4,1f6,7 − 36f2,3f2,5f6,8 + 36f2,3f2,4f6,8 + f4,4f6,3, 0 = −f4,1f6,6 − 36f2,2f2,5f6,8 + 36f2,2f2,4f6,8 + f4,4f6,2, 0 = −3f4,7f6,5 + 108f2,2f2,4f6,5 − 9f2,5f2,4f6,7 + 4f4,5f6,1 − 108f2,1f2,2f6,8 + 8f4,3f6,4, 0 = 96f6,7f 2 2,5 − 99f2,4f6,7f2,5 − f4,7f6,5 + 36f2,2f2,4f6,5 + 4f4,5f6,1 − 36f2,1f2,2f6,8, 0 = 968f4,7f6,6 + 6048f2,2f2,4f6,6 + 1675f4,5f6,7 + 3283f4,8f6,1 − 1056f4,1f6,8 + 2814f4,3f6,3, 0 = 10648f4,7f6,6+66528f2,2f2,4f6,6+4355f4,5f6,7+33768f2,3f2,5f6,7+36113f4,8f6,1−360f4,1f6,8, 0 = 3872f4,7f6,6+24192f2,2f2,4f6,6+1072f4,5f6,7+12663f2,3f2,4f6,7+13132f4,8f6,1−3f4,1f6,8, 0 = f4,5f6,5 − 36f2,3f2,4f6,5 − 6f4,8f6,6 + 36f2,1f2,3f6,8 + 8f4,6f6,2 + f4,2f6,4, 0 = 12f6,6f 2 2,5 + 11f4,5f6,5 − 396f2,3f2,4f6,5 − 71f4,8f6,6 + 396f2,1f2,3f6,8 + 92f4,6f6,2, 0 = 8f4,5f6,5 − 288f2,3f2,4f6,5 − 52f4,8f6,6 + 9f2,5f2,4f6,6 + 288f2,1f2,3f6,8 + 67f4,6f6,2, 0 = 4f4,5f6,6 − 3f4,6f6,8 − 9f2,1f2,4f6,8 + 8f4,1f6,4, 0 = 4f4,5f6,6 − f4,6f6,8 + 96f2,1f2,5f6,8 − 99f2,1f2,4f6,8, 0 = −f4,5f6,5 + 36f2,3f2,4f6,5 − 36f2,1f2,3f6,8 + f4,1f6,3, 0 = 14f4,7f6,6 − 72f2,2f2,4f6,6 + 3f4,1f6,8 + 67f4,2f6,2, 0 = 27f4,7f6,6 + 804f2,2f2,5f6,6 − 828f2,2f2,4f6,6 + f4,1f6,8, 0 = −f4,7f6,5 + 36f2,2f2,4f6,5 − 36f2,1f2,2f6,8 + f4,1f6,2, 0 = −f4,1f6,6 − 48f2,1f2,5f6,7 + 45f2,1f2,4f6,7 + 4f4,3f6,1, 0 = 7f4,6f6,6 − 27f2,1f2,4f6,6 + 2f4,1f6,7 + 26f4,2f6,1, 0 = 27f4,6f6,6 + 624f2,1f2,5f6,6 − 639f2,1f2,4f6,6 + 4f4,1f6,7, 6.4 Automorphic forms, Hecke operators and q-expansions 267 0 = 24f2,5f 2 4,7 + 6f2,3f4,4f4,7 + f4,1f6,5 − 30f2,1f2,2f6,6 + 27f4,2f6,6, 0 = −1260f6,8f32,5 + 1296f2,4f6,8f22,5 − 36f4,8f6,8f2,5 − f6,5f6,1 + f6,2f6,3, 0 = −36f2,3f6,8f22,5 − 11f6,5f6,7 + 180f4,8f2,3f6,8 − 144f2,2f4,4f6,8 + 11f6,8f6,3, 0 = −36f2,2f6,8f22,5 + 144f4,7f6,8f2,5 − 23f6,5f6,6 + 36f2,3f4,4f6,8 + 23f6,8f6,2, 0 = −f26,5 + f26,8 − 36f2,5f4,4f6,8 + 36f2,4f4,4f6,8, 0 = 192f4,8f2,3f6,7 − 405f4,5f2,4f6,7 + 28f6,3f6,7 + 324f4,1f2,4f6,8 − 132f2,1f4,4f6,8 − 7f6,5f6,8, 0 = 768f4,7f2,5f6,6− 2880f4,7f2,4f6,6+189f6,2f6,6− 720f4,1f2,4f6,8+72f2,1f4,4f6,8+7f6,5f6,8, 0 = −4f6,5f6,6 − 87f2,1f4,8f6,7 + 27f2,3f4,3f6,7 + 16f6,7f6,1 + 156f4,7f2,5f6,8, 0 = −4f26,5 + 16f26,7 + 75f2,1f4,5f6,7 − 27f2,5f4,3f6,7 − 60f4,7f2,3f6,8, 0 = −156f2,1f4,8f6,6 + 240f4,6f2,5f6,6 + 27f6,1f6,6 + f6,5f6,7 + 36f4,8f2,3f6,8, 0 = f26,5 + 27f 2 6,6 − 156f2,1f4,7f6,6 + 240f2,2f4,6f6,6 + 36f4,7f2,3f6,8, 0 = 2195416f24,7−13525f24,5−174391f24,8−2887584f24,6−342696f24,1−2016f24,4+70560f2,2f2,3f4,4 + 3528f2,4f6,4, 0 = −181503f2,1f4,8f2,4 + 157248f4,6f2,5f2,4 − 2101f4,7f4,2 − 1072f4,5f4,3 + 416f4,1f4,4 + 24388f2,1f6,4, 0 = −2123163f2,1f4,8f2,4 + 1834560f4,6f2,5f2,4 + 292656f6,1f2,4 − 10665f4,7f4,2 − 2412f4,5f4,3 + 208f4,1f4,4, 0 = 3699f4,7f4,6 + 109824f2,2f2,5f4,6 − 112320f2,2f2,4f4,6 + 184f4,5f4,1 + 441f2,1f4,7f2,4 + 364f2,3f6,5, 0 = 6144f2,5f 2 4,7+1536f2,3f4,4f4,7+256f4,1f6,5+28224f2,1f2,2f6,6+7545f2,1f2,3f6,7−688f4,3f6,7 + 2940f2,2f2,3f6,8 − 84f4,4f6,8 + 147f22,5f6,4, 0 = 17952f2,5f 2 4,7 + 4488f2,3f4,4f4,7 + 748f4,1f6,5 + 82488f2,1f2,2f6,6 + 22080f2,1f2,3f6,7− 2012f4,3f6,7 + 8568f2,2f2,3f6,8 − 245f4,4f6,8 + 441f2,5f2,4f6,4, 0 = 4992f2,5f 2 4,7+1248f2,3f4,4f4,7+208f4,1f6,5+24864f2,1f2,2f6,6+9720f2,1f2,3f6,7−783f4,3f6,7 + 1323f4,8f6,4, 0 = 196f26,5 + 21168f 2 6,4 + 25939968f2,1f4,7f6,6 − 33374208f2,2f4,6f6,6 + 479925f2,1f4,5f6,7 + 941787f2,5f4,3f6,7 − 4328388f4,7f2,3f6,8 + 435456f2,4f4,4f6,8, 0 = 16128f2,2f6,8f 2 2,5 +150492f4,7f6,8f2,5 − 644f6,5f6,6 − 114471f2,1f4,8f6,7 +16491f2,3f4,3f6,7 + 1536f2,3f4,4f6,8 + 2576f6,3f6,4, 0 = −28f26,5 + 112f26,3 − 675f2,1f4,5f6,7 − 717f2,5f4,3f6,7 + 540f4,7f2,3f6,8 − 1536f2,5f4,4f6,8 + 2304f2,4f4,4f6,8, 0 = −6804f2,3f6,8f22,5 + 175296f4,6f6,6f2,5 − 136752f2,1f4,8f6,6 + 77f6,5f6,7 + 18972f4,8f2,3f6,8 + 1296f2,2f4,4f6,8 + 2079f6,2f6,4, 268 Computational applications: kroneckerian polynomials and expansions of automorphic functions and forms 0 = 7f26,5 + 189f 2 6,2 − 12432f2,1f4,7f6,6 + 15936f2,2f4,6f6,6 + 1872f4,7f2,3f6,8 + 324f2,5f4,4f6,8− 972f2,4f4,4f6,8, 0 = −264192f4,7f2,5f6,6 + 1363968f4,7f2,4f6,6 − 28512f4,8f2,3f6,7 + 287955f4,5f2,4f6,7 + 110628f4,1f2,4f6,8 − 1764f2,1f4,4f6,8 + 49f6,5f6,8 + 5292f6,1f6,4, 0 = 192f4,8f2,5f6,7 − 405f4,8f2,4f6,7 + 135f2,1f4,7f6,8 − 108f2,2f4,6f6,8 + 165f2,5f4,2f6,8 − 7f6,5f6,2 + 28f6,1f6,3, 0 = −640f4,7f2,2f6,6 − 2072f4,8f2,5f6,6 − 500f2,1f4,5f6,8 + 240f4,6f2,3f6,8 − 588f2,5f4,3f6,8 + 189f6,1f6,2 + 7f6,5f6,3, 0 = 196f26,5 + 21168f 2 6,1 + 2466816f2,1f4,7f6,6 − 3182592f2,2f4,6f6,6 + 58725f2,1f4,5f6,7 + 134379f2,5f4,3f6,7 − 424836f4,7f2,3f6,8. Resum en català L’objectiu principal d’aquesta tesi és contribuir a la uniformització hiperbòlica explícita de les corbes de Shimura. Ens restringim a les corbes associades a ordres d’Eichler dins d’àlgebres de quaternions racionals tals que el seu quo- cient pel grup d’involucions d’Atkin-Lehner és de gènere 0. Aquest cas, tot i que presenta nombroses diferències amb el cas modular clàssic, també hi té certes similituds. Utilitzem aquest fet per a discutir una aproximació al proble- ma de l’obtenció d’uniformitzacions hiperbòliques explícites d’aquestes corbes i d’alguns recobriments, així com també algunes aplicacions, que il·lustrem amb abundants exemples. Per a entendre millor el problema, començarem introduint breument el seu rerefons històric. Després explicarem en detall les nostres contribucions i el contingut de la memòria. Els grups fuchsians i les funcions automorfes que hi estan associades són un tema d’estudi molt important en les matemàtiques del segle XIX i assoleixen un punt àlgid amb les obres de Poincaré [Poi95], de Klein [Kle23, Kle79], de Fricke [Fri92, Fri93, Fri94, Fri99] i de Fricke i Klein [KF90, KF92, FK97, FK12]. Tot i això, alguns casos particulars de grups fuchsians i funcions automorfes, per exemple les que estan associades a subgrups de congruència Γ0(N) del grup modular Γ0(1) = SL(2,Z), ja es poden veure en les obres de Gauss, Abel o Dedekind. És en aquest context que apareix la funció automorfa més coneguda, la funció j (o invariant j), que també es coneix habitualment com la funció j de Klein, tot i que ja havia estat estudiada abans. Per exemple, Dedekind [Ded77] considera la funció Valenz, que coincideix, llevat d’una constant multiplicativa, amb la funció j, i fins i tot Klein [Kle79] afirma que els orígens d’aquesta funció es remunten a Gauss. La funció j és una funció automorfa per al grup modular SL(2,Z), i per tant, de la mateixa manera que per a tots els subgrups Γ0(N), s’acostuma a anomenar una funció modular (el·líptica). 269 270 Resum en català Si Γ és un grup fuchsià tal que Γ\H té volum hiperbòlic finit (o més gene- ralment, si és un grup fuchsià de primer tipus) i considerem la seva acció en el semiplà superior, aleshores les funcions automorfes respecte Γ són les funcions meromorfes en H que són invariants per l’acció de Γ i que satisfan certes condi- cions de creixement. Per altra banda, l’acció de Γ en H indueix una estructura de superfície de Riemann en l’espai quocient Γ\H que permet identificar Γ\H amb un obert de Zariski del conjunt de punts complexos d’una corba algebrai- ca. En particular, per a alguns dels subgrups de congruència del grup modular, s’obtenen les corbes modulars X0(N). El cos de funcions racionals de la corba modular X0(1) s’identifica natu- ralment amb C(j). Una característi- ca important de la superfície de Rie- mann Γ0(N)\H és que no és compacta. Per exemple, per a N = 1, un domi- ni fonamental per a l’acció de SL(2,Z) en H és el conegut domini fonamen- tal de Gauss, que apareix originalment en el context de la reducció de for- mes quadràtiques binàries. En parti- cular, això permet considerar desenvo- lupaments de Fourier per a les funcions modulars. i ρ i∞ b b a a Per exemple, per a la funció j, els primers termes del desenvolupament de Fourier corresponent són j(τ) = 1 q + 744 + 196884q + 21493760q2 + . . . , on q = e2πiτ . En particular, la racionalitat (i la integralitat) dels coeficients d’aquests desenvolupaments es tradueix en propietats de la corba X0(N). Per exemple, el fet que Γ0(N) = ￿ N 0 0 1 ￿−1 SL(2,Z) ￿ N 0 0 1 ￿ ∩ SL(2,Z) ens permet escriure C(X0(N)) = C(j, j(N ·)). A més a més, existeix un polinomi mònic irreductible i simètric (si N > 1) de grau ψ(N) en cada indeterminada, 271 ΦN (X,Y ) ∈ Z[X,Y ], tal que ΦN (j, j(N ·)) = 0 i que proporciona una equació per a X0(N). Aquest polinomi es coneix habitualment com a equació modular de nivell N . Les arrels del polinomi ΦN (j,X) es poden escriure com j ◦ αN , on αN recorre les transformacions corresponents a un sistema de representants dels elements primitius de norma N en M(2,Z) mòdul SL(2,Z). En aquest cas, a més, és possible escollir els representants αN de manera que fixin ∞, fet que permet obtenir també desenvolupaments de Fourier de les funcions j ◦ αN a partir dels desenvolupaments de j i, en particular, calcular explícitament el polinomi ΦN (X,Y ). A més a més, el polinomi ΦN (X,X) factoritza com a producte d’uns polinomis coneguts com a polinomis de les classes i que tenen com arrels els valors de j en uns punts determinats. D’aquesta manera es poden obtenir explícitament els valors de la funció j en un conjunt de punts del semiplà superior coneguts com a punts de multiplicació complexa (punts CM) i que en aquest cas no són res més que els punts quadràtics de H. A més, el valor de j en un punt CM, τ , no és només un valor algebraic sinó que genera una extensió abeliana del cos Q(τ). En particular, a partir del polinomi ΦN (X,Y ) per a un N adequat, podem obtenir una equació definidora d’aquesta extensió. Per altra banda, també és un resultat clàssic que els punts de la corba X0(1) corresponen a classes d’isomorfisme de corbes el·líptiques generalitzades. Si ex- cloem la punta i∞, un punt τ ∈ SL(2,Z)\H correspon a la classe d’isomorfisme de la corba el·líptica que està representada per C/￿1, τ￿ i els coeficients d’un model de Weierstrass estan donats per mitjà de formes modulars el·líptiques en τ . Aleshores, la funció j es pot escriure com una funció racional dels coeficients del model de Weierstrass. En particular, des d’un punt de vista algebraic, la funció j s’identifica amb un invariant absolut de formes cúbiques (o quàrtiques) binàries que classifica la classe d’isomorfisme de corbes el·líptiques. S’han estudiat moltes més propietats de les funcions modulars; per exemple, la reducció del polinomi de nivell p primer, Φp(X,Y ), mòdul p està donada per la fórmula de congruència de Kronecker; algunes funcions que generen les mateixes extensions abelianes que la funció j en els punts CM però que proporcionen equacions definidores més senzilles, com ara les funcions modulars de Weber, etc. Les formes modulars, com hem vist, també apareixen en aquest context, ja sigui com a funcions theta de Jacobi o intervenint en el desenvolupament de funcions modulars (Eisenstein). Va ser Hecke [Hec24], però, qui va formalitzar-les i també qui va introduir les sèries L associades a formes pròpies i va demostrar algunes de les seves propietats més importants, com ara l’existència de desenvolupaments en forma de producte d’Euler. D’aquesta manera, apareix un concepte clau que ha jugat un paper central en alguns dels resultats més importants de la teoria 272 Resum en català de nombres des de la segona meitat del segle XX. Tot i això, no només s’han estudiat sistemàticament les funcions i formes automorfes associades a subgrups de SL(2,Z). Una de les primeres generalit- zacions que es va considerar és la que correspon al grup modular de Hilbert SL(2,OK), en què Z es substitueix per l’anell d’enters d’un cos de nombres totalment real. D’aquesta manera s’obtenen les funcions i les formes modulars de Hilbert. De la mateixa manera, si substituïm el semiplà superior pel conjunt de les matrius quadrades de dimensió g sobre C amb part imaginària definida positiva i considerem el grup Sp(2g,Z) que hi actua, obtenim les funcions i formes modulars de Siegel, introduïdes per Siegel [Sie35, Sie39] a partir dels treballs de Riemann [Rie57] sobre superfícies de Riemann de gènere g. Per a g = 1, aquesta noció coincideix amb el cas modular clàssic. Per a g = 2, Igusa [Igu60, Igu62, Igu64, Igu67] realitzà un estudi detallat del cos de funcions i de l’anell graduat de formes modulars de Siegel, incloent la seva interpretació com a invariants de formes sèxtiques. Els generadors, en aquest cas, es coneixen com a invariants d’Igusa. És en aquest context, i després dels treballs d’Eichler [Eic37, Eic38, Eic55a, Eic55b] sobre àlgebres de quaternions, que Shimura [Shi67] considera els grups fuchsians obtinguts a través d’una immersió a SL(2,R) dels elements de norma 1 en certs ordres d’una Q-àlgebra de quaternions, H, tal que H⊗QR ∼=M(2,R), introduint així les corbes de Shimura que considerem en aquest treball. Shimura també considera el cas d’àlgebres de quaternions H sobre cossos totalment reals de grau d sobre Q tals que H⊗R ∼=M(2,R)r ×Hd−rR per a un enter 0 < r ≤ d, que dóna lloc a varietats de Shimura de dimensió r. I encara més general- ment, en [Shi75a] la construcció s’estén al cas de grups reductius que actuen en certs dominis. Aquesta construcció fou reformulada per Deligne [Del71, Del79], donant lloc a una reinterpretació que és habitualment utilitzada avui en dia. Nosaltres ens restringirem al cas racional, és a dir, considerarem una àlge- bra de quaternions racional, indefinida i no escindida de discriminant D > 1, HD, i una immersió Φ : HD → M(2,R). Aleshores, com és habitual, fixa- rem un ordre d’Eichler de nivell N , O(D,N) ⊂ HD, i considerarem el grup Γ(D,N) ⊂ PSL(2,R), projecció en PSL(2,R) del subgrup de SL(2,R) obtin- gut a partir dels elements de norma 1 de O(D,N) a través de Φ. Com en el cas modular, un cop fixada la immersió, obtenim uns punts destacats en H, que corresponen a elements fixos per transformacions de HD ￿ Q, que també anomenarem punts CM. És convenient observar que si considerem un punt CM, els elements de HD que el fixen són, de fet, un cos quadràtic K dins HD i, 273 per tant, direm que el punt és CM pel cos K. De fet es pot veure que fixat O(D,N), els punts CM estan naturalment associats a ordres dins de cossos quadràtics mitjançant la noció d’immersió optimal, però, per a introduir les idees generals i explicar de manera resumida els resultats d’aquesta memòria, aquesta identificació menys precisa ja ens serà suficient. Una diferència desta- cable entre aquest cas i el cas modular és que en aquesta situació la superfície de Riemann Γ(D,N)\H corresponent és sempre compacte. En particular, no existeixen desenvolupaments de Fourier per a les funcions automorfes correspo- nents. Gràcies a Shimura [Shi67], sabem que existeix un model canònic per a la superfície de Riemann corresponent a Γ(D,N), és a dir, existeix una corba projectiva X(D,N) definida sobre Q juntament amb un isomorfisme analític jD,N : Γ(D,N)\H → X(D,N)(C), tal que els seus valors en un punt CM per un cos quadràtic K generen determinats cossos de classes d’anell sobre K, en particular, s’obtenen extensions abelianes de K que contenen el cos de classes de Hilbert de K. Per tal de demostrar això, Shimura identifica X(D,N) com un espai groller de moduli. Més precisament, els punts en X(D,N)(C) correspo- nen a classes d’isomorfisme de superfícies abelianes principalment polaritzades amb multiplicació quaterniònica per O(D, 1) (habitualment anomenades corbes el·líptiques falses) i amb una certa estructura de nivell corresponent al nivell N de la corba. Val la pena esmentar que la construcció de Shimura també engloba el cas modular clàssic i la funció jD,N juga el paper de la funció j de Klein. Tot i això, si H ￿∼=M(2,Q), Shimura mateix [Shi75a] demostra que la corba de Shimura corresponent no té punts reals. Per tant, la funció jD,N , en una carta afí, no serà una funció automorfa sinó una tupla de funcions automorfes. També cal destacar l’existència d’un grup d’involucions racionals de la corba X(D,N) que denotem WDN i anomenem grup d’involucions d’Atkin-Lehner. Com a grup abstracte, aquest grup és un producte de grups cíclics d’ordre 2, Cr2 , on r és el nombre de primers diferents que divideixen DN . A més està generat per involucions ωpk on pk￿DN i la seva acció en els punts complexos de la corba correspon a l’acció d’un determinat element de norma pk en O(D,N). En particular, per a cada subgrup W ⊂WDN , obtenim un quocient X(D,N)W de la corba X(D,N) que anomenem quocient d’Atkin-Lehner. Les propietats aritmètiques i diofàntiques de les corbes de Shimura han estat àmpliament estudiades des de la seva introducció als anys 60, per exemple: una uniformització p-àdica va ser obtinguda per Čerednik [Čer76] i Drinfeld [Dri76]; l’estudi de models enters, per Morita [Mor81], Boutot i Carayol [BC91], etc.; l’estudi dels punts locals, per Jordan i Livné [JL85], Ogg [Ogg85], etc.; l’exis- tència de punts racionals en certs quocients d’Atkin-Lehner, per Jordan [Jor81], 274 Resum en català Ogg [Ogg83], etc. A més a més, han tingut un paper important, juntament amb les corbes modulars, en la demostració d’algunes de les conjectures més importants de la teoria de nombres dels últims temps, essencialment a través de la demostració de la modularitat de totes les corbes el·líptiques sobre Q, gràcies als treballs d’Eichler, Shimura i Wiles, entre d’altres. D’aquesta manera, tant les corbes modulars com les de Shimura són espais (grollers) de moduli de corbes el·líptiques (falses) i, al mateix temps, donen parametritzacions modulars per a totes les corbes el·líptiques sobre Q. És convenient fer palès que la funció jD,N no és explícita en la construcció de Shimura, però fins i tot obviant la funció jD,N , determinar equacions per a la corba X(D,N) és en general un problema complicat. La primera equa- ció d’una corba de Shimura (no modular) va ser calculada per Ihara [Iha79] i des de llavors s’han obtingut moltes més equacions seguint diferents tècniques, començant per Kurihara [Kur79], Jordan [Jor81] i Roberts [Rob89] i seguint amb Elkies [Elk98], Gonzàlez i Rotger [GR04, GR06], Molina [Mol12], etc. La majoria d’aquests autors fan ús de la uniformització de Čerednik-Drinfeld per a determinar algunes equacions. Elkies, però, utilitza la uniformització complexa donada pel model canònic de Shimura. D’aquesta manera, recupera també els punts CM corresponents a certs ordres quadràtics. Una combinació de les du- es tècniques també ha estat utilitzada per Yang [Yan13] i Tu [Tu14], estenent d’aquesta manera els casos considerats per Elkies a les corbes que tenen certs recobriments per corbes de gènere 0, afegint el coneixement de certes equacions per a aquests recobriments, essencialment les obtingudes per Gonzàlez i Rotger que acabem de mencionar. Per tal de treballar directament amb formes i funcions automorfes corres- ponents a corbes de Shimura associades a àlgebres de quaternions indefinides i no escindides sobre Q és convenient considerar algun tipus de desenvolupa- ment en sèrie de potències que substitueixi els desenvolupaments de Fourier que s’utilitzen habitualment en el cas modular. Les idees subjacents d’aquests tipus de desenvolupaments es poden remuntar a Shimura [Shi75a] i han estat estudiats des de diferents punts de vista per Mori [Mor94, Mor95, Mor11] i per Bayer [Bay02] i Bayer i Travesa [BT07a, BT07b, BT08]. En particular, en [BT07a, BT07b] es calculen desenvolupaments explícits per a algunes fun- cions modulars i automorfes al voltant de punts CM per a la corba modular i també per a la corba de Shimura associada a un ordre maximal en l’àlgebra de quaternions racional de discriminant 6. El cas de formes automorfes per al cas de discriminant 6 també ha estat estudiat per Baba i Granath [BG12]. Per altra banda, Voight i Willis [VW14] han utilitzat tècniques numèriques per a 275 obtenir desenvolupaments de formes automorfes al voltant de punts CM. De la mateixa manera, Voight [Voi06] ja havia considerat l’ús d’aproximacions per al càlcul de punts CM en el cas triangular. Per altra banda, Voight [Voi10] també considera la determinació de formes automorfes associades a corbes de Shimura mitjançant la correspondència de Jacquet-Langlands. Finalment, el problema de donar equacions per a les corbes de gènere 2 que donen lloc a les superfícies abelianes corresponents als punts de X(D,N)(C) ha estat estudiat per Rotger [Rot04] i Bayer i Guàrdia [BG05] han considerat el seu càlcul explícit utilitzant aproximacions numèriques. Baba i Granath [BG08] obtenen funcions generadores (Hauptmoduln) per al cos de funcions racionals del quocient maximal d’Atkin-Lehner de les corbes de discriminants 6 i 10 mit- jançant invariants d’Igusa a partir de les famílies de Hashimoto-Murabayashi [HM95]. D’aquesta manera, obtenen una fórmula que és similar en essència a la que relaciona j amb els coeficients d’un model de Weierstrass de la corresponent corba el·líptica. En [BT08] aquest Hauptmodul, per a D = 6, es relaciona amb l’obtingut a [BT07b]. Per tant, resulta natural plantejar-se si podem utilitzar els desenvolupaments de funcions automorfes al voltant de punts CM considerats en [BT07b] per ajudar en la determinació d’una uniformització explícita d’un quocient d’Atkin- Lehner de gènere 0 d’una corba de Shimura, contribuint d’aquesta manera a una millor comprensió dels problemes oberts plantejats per Elkies [Elk98, §5.5], en especial aquells referits al càlcul de recobriments i de punts CM. Aquest és el principal tema que hem considerat en aquesta tesi, però, en paral·lel, també tractem altres problemes que ens apareixeran i que comentarem a continuació. El Capítol 1 d’aquesta memòria està dedicat a introduir alguns conceptes i resultats bàsics que resulten necessaris al llarg de la tesi. En el Capítol 2 introduïm la generalització natural de les equacions modulars al nostre context de corbes de Shimura, donant lloc als que denominem polinomis kroneckerians. Per tal de fer això, analitzem algunes de les propietats dels elements de norma N en un ordre d’Eichler O(D,M), amb mcd(DM,N) = 1. Per a construir els polinomis kroneckerians, fixem un subgrup del grup d’in- volucions d’Atkin-Lehner W ⊆ WDM , considerem el grup Γ = Γ(D,M)￿W ￿ i suposem que la corba X(D,M)W és de gènere 0. En particular, existeix un Hauptmodul f : Γ\H→ C ∪ {∞}. Aleshores, veiem que el polinomi Ψ˜Γ,N,f (X, f) = ￿ γ∈(Γ∩ω−1N ΓωN )\Γ (X − f ◦ ωN ◦ γ), 276 Resum en català on ωN és la transformació associada a un element primitiu de norma N en O(D,M) que representa la involució d’Atkin-Lehner de nivell N en X(D,MN), pertany a C(f)[X]. Definició. Anomenem polinomi kroneckerià de nivell N associat al Hauptmo- dul f pel grup Γ a un polinomi ΨΓ,N,f (X,Y ) ∈ C[X,Y ] primitiu com a polinomi en la variable Y i tal que ΨΓ,N,f (X, f) = P (f)Ψ˜Γ,N,f (X, f), on P (f) ∈ C(f). En particular, convé notar que aquests polinomis estan determinats úni- cament mòdul una constant multiplicativa de C∗. Les seves propietats més bàsiques estan recollides en el Teorema 2.1. Teorema 2.1. Sigui f un Hauptmodul per a la corba X(D,M)WC . Per a tot enter positiu N coprimer amb DM , el polinomi ΨΓ,N,f (X,Y ) és un polinomi irreductible de grau ψ(N) en cada indeterminada. Si N = 1, podem pren- dre ΨΓ,1,f (X,Y ) = X − Y ; si N > 1, aleshores ΨΓ,N,f (X,Y ) és simètric. A més a més, si f és un Hauptmodul definit sobre un cos de nombres K, ales- hores, per a una elecció adequada de la constant multiplicativa de la definició, ΨΓ,N,f (X,Y ) ∈ K[X,Y ]. En particular, el polinomi ΨΓ,N,f (X,Y ) proporciona un model pla (i singu- lar) per a la corba X(D,MN)W sobre K, si el Hauptmodul està definit sobre K, però no només això ja que també podem obtenir com a corol·lari un model per a la corba X(D,MN)￿W,ωN ￿ sobre K. Corol·lari 2.3. Siguin K un cos de nombres i f un Hauptmodul definit sobre K de X(D,M)W . Per a tot enter N > 1 coprimer amb DM , el polinomi ΨΓ,N,f (X + √ Y ,X −√Y ) pertany a K[X,Y ], és irreductible sobre C i proporciona un model sobre K per a la corba X(D,MN)￿W,ωN ￿. Com en el cas clàssic, en la Proposició 2.5 demostrem que, a partir dels polinomis kroneckerians, també podem obtenir els valors del Hauptmodul cor- responents a uns punts CM determinats. Proposició 2.5. Siguin W un subgrup de WDM , Γ = Γ(D,M)￿W ￿ i f un Hauptmodul per la corba X(D,M)WC . Considerem el conjunt S = {s : ωs ∈W}. 277 Donat τ0 ∈ H un punt CM per l’ordre Λ en el cos quadràtic K, la multipli- citat f(τ0) com a arrel de ΨΓ,N,f (X,X) és donada per r(Λ, N, S) = #{λ ∈ Λ￿ : ωsφ(λ) ∈ sO(D,M)N per a un cert s ∈ S}/ ∼ = #{λ ∈ Λ￿ : ωsφ(λ) ∈ sO(D,M)N per a un cert s ∈ S}/2e(τ0), on x ∼ y si i només si xy ∈ NΛ, e(τ0) denota l’ordre el·líptic del punt τ0 i · denota la conjugació. En particular, z0 ∈ C és una arrel del polinomi ΨΓ,N,f (X,X) ∈ C[X] si i només si z0 = f(τ0) per a un punt CM, τ0 ∈ H, associat a una immersió optimal φ : Λ ￿→ O(D,M) tal que existeix un element primitiu λ ∈ Λ de norma sN per a un cert s ∈ S per al qual ωsφ(λ) ∈ sO(D,M) (cf. Proposició 2.3). Finalment, pel que fa a les propietats generals dels polinomis kroneckerians, ens centrem a estudiar les seves singularitats. Comencem observant que totes les seves singularitats provenen de punts CM i aleshores ens concentrem en l’estudi del nombre de branques per un punt CM determinat. Més precisament, anome- nem multiplicitat de Ψ en (P,ωNP ), i la denotem mult(P,ωNP )Ψ, al nombre de branques de Ψ(X,Y ) = 0 a través de (f(P ), f(ωNP )). En el Teorema 2.2 relacionem aquesta multiplicitat amb un determinat ordre d’anul·lació. Teorema 2.2. Sigui X(D,M)W una corba de gènere 0 amb un Hauptmodul f . Denotem per O i Γ = ￿Γ(D,M),W ￿ l’ordre i el grup corresponents. Per a un enter positiu N coprimer amb DM , fixem un ordre d’Eichler de nivell MN dins O, O(N), i considerem Γ(N) = ￿Γ(D,MN),W ￿ ⊂ Γ i ωN la involució de nivell N corresponent. Per a un punt P ∈ H, denotem per ΓP = {γ ∈ Γ : γ(P ) = P} la isotropia de P i (Γ\O￿N )P = {Γα ∈ Γ\O￿N : α(P ) = ωN (P ) en Γ\H}. Aleshores, (1) (ΓP ∩ Γ(N))\ΓP actua per la dreta en (Γ\O￿N )P , de la manera següent ·[￿] : (Γ\O￿N )P → (Γ\O￿N )P Γα ￿→ Γα￿. A més a més, aquesta acció és lliure, és a dir, si [￿] ￿= [1], llavors ·[￿] no fixa cap element. 278 Resum en català (2) Les branques del polinomi Ψ(X,Y ) = ΨΓ,N,f (X,Y ) a través de (P,ωNP ) estan en correspondència bijectiva amb les òrbites d’aquesta acció. Els ele- ments en una òrbita fixada, αi, donen els elements (f, f ◦αi) que defineixen una branca. En particular, ordf(ωNP )Ψ(f(P ), X) = e1(P ) eN (P ) mult(P,ωN (P ))Ψ(X,Y ), on ea(P ) denota l’ordre el·líptic del punt P en X(D,Ma)W . En el Teorema 2.3, ens centrem en el cas de polinomis kroneckerians de nivell potència de primer i donem una descripció explícita dels punts singulars i de la seva multiplicitat. En la segona part del Capítol 2 ens centrem en les corbes X(D,N) de gè- nere 0, que són exactament X(D, 1), D = 6, 10, 22. En el Teorema 2.4 donem una construcció unificada dels Hauptmoduln tD per a aquestes tres corbes i uD,1, uD,2, uD,3 i t+D per als seus quocients d’Atkin-Lehner. Els Hauptmoduln tD estan definits sobre cossos quàrtics i la resta sobre Q. Tot i que tD no estigui definit sobre Q, en la Proposició 2.6 provem que els polinomis kroneckerians corresponents sí que estan definits sobre Q. Proposició 2.6. Per a tot N > 1 coprimer amb D, el polinomi ΨΓ,N,tD (X,Y ) es pot prendre com un polinomi primitiu en Z[X,Y ]. A més, el grau global de tots els monomis no nuls de ΨΓ,N,tD (X,Y ) és parell i, si escrivim ΨΓ,N,tD (X,Y ) = ψ(N)￿ j,k=0 ajkX jY k, aj,k ∈ Z, aleshores se satisfan les relacions següents: ajk = akj = aψ(N)−k,ψ(N)−j = aψ(N)−j,ψ(N)−k. Finalment, en el Teorema 2.5 estudiem la reducció mòdul p dels polinomis ΨΓ,p,tD (X,Y ) a partir de la llei de reciprocitat de Shimura i de les relacions concretes entre les funcions uD,1, uD,2, uD,3. Obtenim així un anàleg en aquest cas de la fórmula de congruència de Kronecker. Teorema 2.5. Siguin D = 6, 10, 22 i p un primer que no divideixi D. Aleshores, la forma general de la reducció mòdul p del polinomi kroneckerià ΨΓ,p,tD (X,Y ) 279 només depèn de p (mod 4c(D)). Més precisament, llevat d’una constant multi- plicativa no nul·la, es té que ΨΓ,p,tD (X,Y ) ≡  (X − Y p)(Y −Xp) (mod p), si p ≡ 1 (mod4) i ￿ c(D) p ￿ = 1, (X + Y p)(Y +Xp) (mod p), si p ≡ 1 (mod4) i ￿ c(D) p ￿ = −1, (1−XY p)(1− Y Xp) (mod p), si p ≡ 3 (mod4) i ￿ c(D) p ￿ = −1, (1 +XY p)(1 + Y Xp) (mod p), si p ≡ 3 (mod4) i ￿ c(D) p ￿ = 1, on c(D) =  3, si D = 6, 2, si D = 10, 11, si D = 22. En el Capítol 3 comencem introduint l’algoritme que hem utilitzat per a calcular dominis fonamentals per als grups Γ(D,N) basant-nos en el mètode de Ford i l’utilitzem per al càlcul de dominis fonamentals per als grups Γ(22, 1) i Γ(D, 1)￿WD￿ per a D = 6, 10, 22. A continuació, introduïm els paràmetres res- pecte els quals farem els desenvolupaments al voltant d’un punt de multiplicació complexa P , q(z) = κ z−P z−P , κ ∈ C∗, i les equacions diferencials satisfetes pels Hauptmoduln: donat un Hauptmodul f : H → C ∪ {∞} corresponent a una corba de gènere 0, aleshores existeix una funció racional R(X) ∈ C(X) tal que Da(f(z), z) +R(f(z)) = 0, on Da(f(z), z) = 2f ￿(z)f ￿￿￿(z)− 3(f ￿￿(z))2 (f ￿(z))4 i f ￿(z) = df(z)dz . La Proposició 3.8 dóna una descripció més explícita de la funció R, però llevat d’algun cas molt concret, no se’n té una expressió concreta en ge- neral. També estudiem la relació d’aquesta equació amb equacions diferencials lineals. Llavors, utilitzem aquests desenvolupaments, l’equació diferencial satis- feta per un Hauptmodul fixat i un polinomi kroneckerià corresponent de nivell petit per a determinar una uniformització explícita de la corba X(22, 1)W22 , que pel que sabem, no havia estat calculada abans. També recuperem les uniformit- zacions de les corbes X(D, 1)WD , per a D = 6, 10, obtingudes amb anterioritat per Elkies [Elk98] i Bayer i Travesa [BT07b]. Resumim totes aquestes unifor- mitzacions en el Teorema 3.1. 280 Resum en català Teorema 3.1. Sigui D = 6, 10, 22. (1) Sigui f qualsevol de les funcions introduïdes en el Teorema 2.4. Aleshores, existeix una funció racional R(X) ∈ Q[X] tal que Da(f(z), z)+R(f(z)) = 0 i el valor exacte de R(X) es pot trobar en la taula següent: f D 6 10 t+D 108−113f+140f2 144(−1+f)2f2 2187−3183f+3067f2−208f3+12f4 4(−1+f)2f2(−27+2f)2 uD,1 −113+9f2 12(3+f2)2 − 10(303−22f 2+7f4) (2+f2)2(27+f2)2 uD,2 96−49f2+9f4 12f2(3+f2)2 2(−1075+614f2+29f4) (−5+f)2(5+f)2(2+f2)2 uD,3 −103+32f2 36(1+f2)2 − 2859+5522f2+2675f4 (1+f2)2(27+25f2)2 tD 27+74f2+27f4 36(−1+f)2f2(1+f)2 128(1+23f2+f4) (1−52f2+f4)2 f D 22 t+D 8748−17697f+18316f2−10624f3+3072f4 16(−1+f)2f2(−27+16f)2 uD,1 −80619−6743f2−193f4+3f6 4(11+f2)2(27+f2)2 uD,2 −2816+8800f2+61f4+3f6 4(−4+f)2(4+f)2(11+f2)2 uD,3 − 10167+14240f2+5888f44(1+f2)2(27+16f2)2 tD 11(33+508f2+3014f4+508f6+33f8) 4f2(11−86f2+11f4)2 (2) Els valors de les funcions t+D en els punts CM per l’anell d’enters d’alguns cossos quadràtics imaginaris de discriminants petits es poden trobar a la taula següent. El símbol ∗ indica que no hi ha punts CM per aquest cos en 281 la corba corresponent. −3 −4 −8 −11 −19 −20 −24 −40 t+6 ∞ 1 ∗ ∗ 32111024 ∗ 0 2312125 t+10 27 2 ∗ 1 ∗ ∗ ∞ ∗ 0 t+22 27 16 1 ∗ ∞ ∗ 516 ∗ ∗ Aleshores, busquem una expressió del Hauptmodul t+22 en funció dels in- variants d’Igusa. Per tal de fer això considerem una immersió de la corba X(22, 1)W22 en la varietat d’Igusa i identifiquem la imatge com un cert fac- tor d’una intersecció de superfícies de Humbert. Això ens permet obtenir una expressió per al Hauptmodul t+22 mitjançant invariants d’Igusa tal com recollim en el Teorema 3.10. Teorema 3.10. Siguin ￿22 : H→ H2 la immersió de H en el semiplà superior de Siegel de gènere 2 donada a la pàgina 96, i g22 : H→ C∪ {∞} definida com g22(z) = g22,1 g22,2 , on g22,1 =11(2392824775436786785s2(￿22(z)) 4s3(￿22(z)) − 30647593176686406000s2(￿22(z))s3(￿22(z))3 − 3582508007923968630s2(￿22(z))3s5(￿22(z)) + 79794269444084577696s3(￿22(z)) 2s5(￿22(z)) − 59164507364503662000s2(￿22(z))s3(￿22(z))s6(￿22(z)) + 115473232412932391712s5(￿22(z))s6(￿22(z))), g22,2 =2528s2(￿22(z))(211603945426835s2(￿22(z)) 3s3(￿22(z)) − 114771506545713600s3(￿22(z))3 − 106795942633963098s2(￿22(z))2s5(￿22(z)) + 613643668464340800s3(￿22(z))s6(￿22(z))), 282 Resum en català i s2 =12(J 2 2 − 24J4), s3 =12(J 3 2 − 36J2J4 + 216J6), s5 =60(82944J10 + J 5 2 − 60J32J4 + 864J2J24 + 216J22J6 − 5184J4J6), s6 =12(497664J10J2 + 11J 6 2 − 792J42J4 + 18144J22J24 − 124416J34 + 864J32J6 − 31104J2J4J6 + 93312J26 ), on J2, J4, J6, J10 són els invariants d’Igusa projectius. Aleshores, el Hauptmodul t+22 admet l’expressió següent: t + 22 = 925g22 + 176 1024g22 + 176 . Llavors, utilitzant el mètode de Mestre, podem recuperar equacions per a una corba que dóna lloc a la superfície abeliana en funció de t+22 o, equivalentment, en funció de u22,2, u22,3. Per a poder fer això, hem d’eliminar un conjunt finit de punts que també determinem explícitament, tal com apareix en el Corol·lari 3.4. Corol·lari 3.4. Sigui S el conjunt de punts CM associats als anells d’enters dels cossos quadràtics K ∈ {Q(i),Q(√−3),Q(√−5),Q(√−11)}. Les funcions v1 = u22,3 i v2 = 1u22,2 satisfan v 2 1 + 11v 2 2 + 1 = 0 i donen una altra carta afí per al model canònic de X(22, 1) donat al Teorema 2.4. Aleshores, una corba de gènere 2 que dóna lloc a la superfície abeliana corresponent a un punt de X(22, 1)(C)￿ S és donada per Y 2 =− 512√2X ￿3− 2X2 + 3X4￿ v51 + 256 √−22(−1 +X)(1 +X) ￿1 + 10X2 +X4￿ v41v2 + 512(1 +X2)3v41 − 2048√−11(1 +X2)(−1 +X)X(1 +X)v31v2 − 64√2X ￿57− 134X2 + 57X4￿ v31 + 96 √−22(−1 +X)(1 +X) ￿1 + 42X2 +X4￿ v21v2 + 208 ￿ 1 +X2 ￿ ￿ 1 + 26X2 +X4 ￿ v21 − 640√−11(1 +X2)(−1 +X)X(1 +X)v1v2 − 2√2X ￿759− 4282X2 + 759X4￿ v1 + 5 √−22(−1 +X)(1 +X) ￿1 + 202X2 +X4￿ v2 − 7 ￿1 +X2￿ ￿1− 598X2 +X4￿ . 283 Finalment, també estudiem la bona reducció potencial de les corbes obtin- gudes. Proposició 3.19. Donat un punt de X(22, 1)+(Q), la corba corresponent té bona reducció potencial en p ￿= 2, 11 si i només si νp((t + 22 − 1)(16t+22 − 27)(16t+22 − 5)) = 0. En el Capítol 4 comencem calculant totes les parelles (D,N) tals que la cor- ba X(D,N)WDN té gènere 0, que és el conjunt de corbes per a les quals podríem seguir el mateix mètode que hem utilitzat en el Capítol 3 per a obtenir-ne una uniformització explícita, almenys des d’un punt de vista teòric. En la Propo- sició 4.1, donem una llista de les 271 corbes X(D,N) que tenen un quocient d’Atkin-Lehner de gènere 0 juntament amb tota la informació necessària per a calcular el gènere i el nombre de cicles el·líptics (i els corresponents ordres quadràtics) de qualsevol dels seus quocients d’Atkin-Lehner. Aquelles corbes amb N lliure de quadrats ja han estat determinades a [LMR06]. A més a més, demostrem que en tots aquests casos, la corba X(D,N)WDN és isomorfa a P1Q. També podem observar directament que el nombre de cicles el·líptics de la corba X(D,N)WDN és sempre més gran o igual que el de la corba X(D, 1)WD , que també és de gènere 0. Com que la complexitat computacional del procediment descrit en el Capítol 3 depèn en gran part del nombre de cicles el·líptics, és mi- llor determinar una uniformització explícita de la corba X(D, 1)WD i utilitzar-la després per a obtenir una uniformització de X(D,N)WDN . En aquest capítol, veiem com podem combinar els polinomis kroneckerians amb les idees utilitzades en [Elk98] per a fer-ho. Per a il·lustrar aquest procediment, considerem les cor- bes estudiades en el Capítol 3 i uniformitzem les corbes X(D,N), D = 6, 10, 22, juntament amb els seus quocients, per als tres valors més petits possibles de N > 1 en cada cas. Donem Hauptmoduln per als quocients racionals i equa- cions per a la resta de corbes de gèneres 0, 1, 2 o hiperel·líptiques de gènere superior. En la Secció 4.2 estudiem el cas (22, 3) en detall i en les seccions posteriors estudiem la resta de casos, on ja posem l’èmfasi en les particularitats de cada cas. Pel que sabem, no es tenien uniformitzacions explícites de les cor- bes corresponents a (6, 11), (10, 9) ni per a cap de les de discriminant 22. En aquest cas la complexitat del càlcul depèn en gran part del grau del recobriment X(D,N)WD → X(D, 1)WD , que és ψ(N). Ara bé, si N és compost (o potència de primer), podem trencar aquest càlcul en parts més petites (per exemple per a X(10, 9)). A més a més, també podem utilitzar aproximacions numèriques o 284 Resum en català p-àdiques, i fins i tot podem utilitzar programari d’àlgebra computacional per a resoldre els casos més senzills. En el Capítol 5 estudiem les propietats aritmètiques dels coeficients dels desenvolupaments, respecte paràmetres locals adequats, de les formes (i funci- ons) automorfes associades als grups Γ(D,N) al voltant dels punts CM. Per tal de fer això, donats un punt CM, P , i una forma automorfa de pes 2r, f , considerem un paràmetre local en P de la forma q(z) = κ z−P z−P amb κ ∈ C∗ i aleshores el corresponent desenvolupament f = 1 (2πi)r ￿ (q − κ)2 κ(P − P ) ￿r￿ n≥0 anq n ∈ A2r(Γ). En primer lloc, relacionem els coeficients an amb els valors de certs operadors diferencials, íntimament relacionats amb els operadors de Shimura-Maaß. Ales- hores, procedint de manera similar a [Shi75a], obtenim un resultat d’algebrai- citat per als coeficients an, després de normalitzar-los adequadament, formulat sobre cossos de nombres explícits, que enunciem en el Teorema 5.1. En parti- cular, demostrem l’algebraicitat dels coeficients d’aquests desenvolupaments de funcions automorfes per Γ(D,N), generalitzant així a D arbitrari i sobre cossos de nombres explícits els corresponents resultats donats en [BT08]. Corol·lari 5.2. Sigui P un punt CM per un ordre Λ ⊂ OK . Denotem KΛ el corresponent cos de classes d’anell. Aleshores, existeix κ = κP ∈ C∗ tal que per a tot f ∈ A2r(Γ)L·KΛ , an ∈ L ·KΛ, per a tot n. A més a més, κπ−1DK ∈ Q, per a πDK = |DK |￿ m=1 Γ ￿ m |DK | ￿￿DK m ￿s , on s =  2, si K = Q(i), 3, si K = Q( √−3), 1/h(K), altrament. En el Teorema 5.2 demostrem en certa manera l’altra implicació, donant una caracterització del cos de definició d’una forma o funció a partir dels coeficients del desenvolupament al voltant d’un punt CM. 285 Teorema 5.2. Siguin P un punt CM per Λ ⊂ OK , κ ∈ C∗ com en el corol·lari anterior i q el corresponent paràmetre local. Sigui també f ∈ A2r(Γ). Aleshores, f ∈ A2r(Γ)L·KΛ si i només si an ∈ L ·KΛ per a tot n. A partir d’aquest resultat, en el Corol·lari 5.3 obtenim una nova prova d’un resultat d’Ihara [Iha74] que ens assegura que els coeficients de l’equació diferen- cial corresponent a un Hauptmodul definit sobre un cos també té coeficients en el mateix cos. Corol·lari 5.3. Sigui f ∈ A0(Γ)L. Aleshores, Da(f, z) ∈ A0(Γ)L. Finalment, en el Teorema 5.3, estudiem l’acció del grup de Galois en els coeficients d’aquests desenvolupaments i obtenim d’aquesta manera una extensió de la llei de reciprocitat de Shimura a tots els coeficients del desenvolupaments d’una funció i no només per als valors de la funció, que podem interpretar com la llei de reciprocitat de Shimura clàssica. En el Capítol 6 tractem algunes aplicacions més computacionals de les eines i els objectes introduïts en els capítols previs. En la Secció 6.1 estudiem el càlcul explícit de polinomis kroneckerians d’un determinat nivell quan es tenen desenvolupaments al voltant de certs punts CM i a més a més donem alguns exemples per a D = 6. Tot i això, per a algunes de les aplicacions posteriors, hem calculat alguns polinomis més que per motius d’espai no hem reproduït, més precisament per a D = 6, 10, 22 hem calculat els polinomis kroneckerians de nivell primer per a tots els primers coprimers amb D i menors o iguals que 41. En la Secció 6.2 considerem el problema de calcular els valors d’un Hauptmo- dul donat en un punt CM concret de manera determinista, sense fer ús d’apro- ximacions, gràcies a la Proposició 2.5. En particular, ho utilitzem per a calcular tots els punts CM racionals i quadràtics de les corbes X(D, 1)WD , D = 6, 10, 22, en la Secció 6.2.1. Així obtenim una nova demostració que els valors racionals calculats per Elkies a [Elk98] per als casosD = 6, 10 són correctes (també donem un valor omès en [Elk98] per al cas D = 10). Una altra demostració utilitzant eines completament diferents va ser obtinguda per Errthum [Err11]. El nostre procediment també es pot utilitzar per a comprovar si els valors de les taules en [Elk98] per als casos D = 14, 15 són correctes. A més a més, no ens restringim al cas dels valors dels Hauptmoduln i també proporcionem un mètode per al càlcul dels primers termes d’un desenvolupament en un punt CM arbitrari. Un cop es disposa dels primers termes, es pot utilitzar l’equació diferencial per a calcular tants termes com calguin. 286 Resum en català En la Secció 6.3 comencem donant els primers termes dels desenvolupaments al voltant de tots els punts CM racionals per als Hauptmoduln de cadascuna de les corbes X(D, 1)WD que hem determinat anteriorment. Tot i això, per a poder treballar realment amb aquestes funcions no només calen els desenvolupaments, sinó també el paràmetre q respecte al qual hem calculat el desenvolupament; també donem aquests valors en la Secció 6.3. Cal destacar, però, que aques- tes constants, amb l’excepció de tres valors per al cas D = 6, s’han obtingut mitjançant aproximació i identificació i per tant, tot i que creiem fermament que els valors que donem són correctes, com que no coneixem cap mètode per a demostrar que ho són, només podem garantir que són aproximacions dels valors reals (com a mínim fins a 150 xifres decimals, però en cada cas es poden com- provar més dígits). En qualsevol cas, això és suficient, en general, per a obtenir aproximacions dels valors de la funció o de la seva derivada en un entorn del punt. Finalment, en la Secció 6.4 tractem el problema de calcular desenvolupa- ments per a formes automorfes. Amb aquest propòsit, suposem que X(D,N) és tal que X(D,N)WDN és de gènere 0 i en la Proposició 6.5 calculem una base de l’espai de formes de pes k per a la corba X(D,N) a partir d’un Hauptmodul per a X(D,N)WDN . 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