Borcherds products in the arithmetic intersection theory of Hilbert modular surfaces

We prove an arithmetic version of a theorem of Hirzebruch and Zagier saying that Hirzebruch-Zagier divisors on a Hilbert modular surface are the coefficients of an elliptic modular form of weight two. Moreover, we determine the arithmetic self-intersection number of the line bundle of modular forms equipped with its Petersson metric on a regular model of a Hilbert modular surface, and study Faltings heights of arithmetic Hirzebruch-Zagier divisors.


Introduction
It is of special interest in Arakelov theory to determine intrinsic arithmetic intersection numbers of varieties defined over number fields and Faltings heights of its subvarieties. Here, we study Hilbert modular surfaces associated to a real quadratic field K. Since these Shimura varieties are noncompact, we consider suitable toroidal compactifications. As the natural metrics of automorphic vector bundles on such varieties have singularities along the boundary (see, e.g., [M], [BKK1]), we work with the extended arithmetic Chow rings constructed in [BKK2].
In their celebrated article [HZ], Hirzebruch and Zagier proved that the generating series for the cohomology classes of Hirzebruch-Zagier divisors is a holomorphic modular form of weight 2. A different proof was obtained by Borcherds [B2]. Moreover, it was shown by Franke [F] and by Hausmann [Ha] that the product of this generating series with the first Chern class of the line bundle of modular forms (i.e., the generating series for the hyperbolic volumes) equals a particular holomorphic Eisenstein series E(τ ) of weight 2.
In the present article, we define certain arithmetic Hirzebruch-Zagier divisors on a suitable regular model of the Hilbert modular surface. We show that the generating series of their classes in the arithmetic Chow ring is a holomorphic modular form (of the same level, weight, and character as in the case of Hirzebruch and Zagier). The main result of our work is that the product of this generating series with the square of the first arithmetic Chern class of the line bundle of modular forms equipped with its Petersson metric is equal to a multiple of E(τ ). The factor of proportionality is the arithmetic self-intersection number of the line bundle of modular forms which is computed explicitly.
Since Hilbert modular surfaces can also be viewed as Shimura varieties associated to the orthogonal group of a rational quadratic space of signature (2, 2), these results are related to the program described by Kudla in [Ku2], [Ku3], [Ku4], [Ku6] (for a discussion of results in that direction, see also the references therein). However, notice that there are several technical differences. For instance, in the present case, the Shimura variety has to be compactified, the arithmetic Hirzebruch-Zagier divisors contain boundary components, and we work with different Green functions. We show that the arithmetic self-intersection number of the line bundle of modular forms is essentially given by the logarithmic derivative at s = −1 of the Dedekind zeta function ζ K (s) of K. It equals the expression conjectured in [Kü1]. We refer to Theorem B for the precise statement.
We also determine the Faltings heights of those Hirzebruch-Zagier divisors that are disjoint to the boundary. It is well known that their normalizations are isomorphic to compact Shimura curves associated with quaternion algebras. This result will be used in a subsequent work [KK] to determine the arithmetic self-intersection number of the Hodge bundle equipped with the hyperbolic metric on Shimura curves.
Our formulas provide further evidence for the conjecture of Kramer, based on results obtained in [K] and [Kü1], that the arithmetic volume is essentially the derivative of the zeta value for the volume of the fundamental domain, the conjecture of Kudla on the constant term of the derivative of certain Eisenstein series (see [Ku3], [Ku4], [Ku6], [KRY]), and the conjecture of Maillot and Roessler on special values of logarithmic derivatives of Artin L-functions (see [MR1], [MR2]).
Our approach requires various results on regular models for Hilbert modular surfaces (see [R], [DP], [P]), as well as an extensive use of the theory of Borcherds products (see [B1], [B2], [Br1], [Br2]). Another central point is the q-expansion principle (see [R], [C]), which relates analysis to geometry. Finally, a result on Galois representations (see [OS]) allows us to replace delicate calculations of finite intersection numbers with density results for Borcherds products.
We now describe the content of this article in more detail.
Since the natural metrics on automorphic line bundles have singularities along the boundary, we cannot use arithmetic intersection theory as presented in [SABK] and have to work with the extension developed in [BKK2]. In particular, our arithmetic cycles are classes in the generalized arithmetic Chow ring CH * (X , D pre ), in which the differential forms are allowed to have certain log-log singularities along a fixed normal crossing divisor. We recall some of its basic properties in Section 1. Choosing representatives, the arithmetic intersection product of two arithmetic cycles can be split into the sum of a geometric contribution as in [SABK] and an integral over a star product of Green objects. The formulas for the latter quantity generalize those of [SABK] and may contain additional boundary terms. We concentrate on the analytic aspects in Sections 2 and 3. The geometric contribution is considered in Sections 5 and 6.
We begin Section 2 by recalling some facts on the analytic theory of Hilbert modular surfaces. Throughout the article, the discriminant D of K is assumed to be prime. Let be a subgroup of finite index of the Hilbert modular group K = SL 2 (O K ), where O K denotes the ring of integers in K. We consider an (at the outset, arbitrary) desingularization X( ) of the Baily-Borel compactification X( ) of \H 2 , where H is the upper complex half-plane. Here, the "curve lemma" due to Freitag [Fr1, Satz 1, Hilfssatz 2] is a main technical tool to obtain adequate local descriptions. We introduce the line bundle of modular forms M k ( ) of weight k with its Petersson metric · Pet , which is singular along the normal crossing divisor formed by the exceptional curves of the desingularization. We show that the Petersson metric is a pre-log singular hermitian metric in the sense of [BKK2].
For any positive integer m, there is an algebraic divisor T (m) on X( ) called the Hirzebruch-Zagier divisor of discriminant m. These divisors play a central role in the study of the geometry of Hilbert modular surfaces (see, e.g., [G]). We consider the Green function m (z 1 , z 2 , s) on the quasi-projective variety \H 2 , associated to T (m), introduced in [Br1]. A new point here is the explicit description of the constant term, given in Theorem 2.11, which involves ζ K (s) and a certain generalized divisor function σ m (s) (see (2.32)). We then define the automorphic Green function G m for T (m) to be the constant term in the Laurent expansion at s = 1 of m (z 1 , z 2 , s) minus a natural normalizing constant, which is needed to obtain compatibility with the theory of Borcherds products as used in Section 4. By abuse of notation, we also denote by T (m) the pullback of T (m) to X( ). By means of the curve lemma, we show that the Green function G m defines a pre-log-log Green object in the sense of [BKK2] for the divisor T (m) in the projective variety X( ).
In Section 3, we specialize the general formula for the star product to the Green objects g(m j ) = (−2∂∂G m j , G m j ) associated to suitable triples of Hirzebruch-Zagier divisors T (m j ) (see Theorem 3.3). In this case, using the curve lemma, it can be shown that certain boundary terms vanish. As a consequence, the resulting formula depends only on the Baily-Borel compactification X( ) and is independent of the choices made in the desingularization. An analogous formula holds for the star product associated to a Hirzebruch-Zagier divisor T (m) and the divisors of two Hilbert modular forms.
In that way, we reduce the analytical contributions to the arithmetic intersection numbers in question to integrals of Green functions G m over X( ) and star products on Hirzebruch-Zagier divisors T (m). The analysis of Section 2 allows us to determine the integral of G m in terms of the logarithmic derivatives of ζ K (s) and σ m (s) at s = −1 (Corollary 3.9). If p is a prime that splits in K, then T (p) is birational to the modular curve X 0 (p). We determine the star products on these T (p) using the results of [Kü2] in Theorem 3.13.
The proofs of our main results rely on the theory of Borcherds products in a vital way. Therefore, in Section 4, we recall some of their basic properties with an emphasis on the construction given in [Br1]. Borcherds products on Hilbert modular surfaces are particular meromorphic Hilbert modular forms that arise as regularized theta lifts of certain weakly holomorphic elliptic modular forms of weight zero. They enjoy striking arithmetic properties. For instance, a sufficiently large power of any holomorphic Borcherds product has coprime integral Fourier coefficients. Hence, by the q-expansion principle, it defines a section of the line bundle of modular forms over Z. Moreover, the divisors of Borcherds products are explicit linear combinations of Hirzebruch-Zagier divisors, dictated by the poles of the input modular form. By [Br1], the logarithm of their Petersson norm is precisely given by a linear combination of the Green functions G m .
By a careful analysis of the obstruction space for the existence of Borcherds products, we prove that for any given linear combination of Hirzebruch-Zagier divisors C, there are infinitely many Borcherds products F 1 , F 2 such that C ∩ div (F 1 ) ∩ div (F 2 ) = ∅ and such that further technical conditions are satisfied (Theorem 4.12). This gives an ample supply of sections of the line bundle of modular forms for which the associated star products can be calculated. In the rest of this section, we essentially show that the subspace of Pic(X( K )) ⊗ Z Q, spanned by all Hirzebruch-Zagier divisors, is already generated by the T (p), where p is a prime that splits in K (Corollary 4.16). This fact can be viewed as an explicit moving lemma for Hirzebruch-Zagier divisors (see Theorem 6.1), and it is crucial in the proof of Theorem B since it noticeably simplifies the calculations at finite places.
In Section 5, we recall the arithmetic theory of Hilbert modular surfaces. Unfortunately, there are currently no references for projective regular models defined over Spec Z. Therefore we work with regular models over the subring Z[ζ N , 1/N] of the Nth cyclotomic field Q(ζ N ). More precisely, we consider toroidal compactifications H(N) of the moduli scheme associated with the Hilbert modular variety for the principal congruence subgroup K (N) of arbitrary level N ≥ 3. We define T N (m) ⊆ H(N) as the Zariski closure of the Hirzebruch-Zagier divisor T (m) on the generic fiber. This definition is well behaved with respect to pullbacks and is compatible with the theory of Borcherds products (see Proposition 5.7). If p is a split prime, there exists a natural modular morphism from the compactified moduli space of elliptic curves with a subgroup of order p to the minimal compactification of H(1) whose image is essentially T 1 (p) (see Proposition 5.13). This allows us to determine the geometric contribution of the arithmetic intersection numbers in question using the projection formula (Proposition 5.15).
In Section 6, the arithmetic intersection theory of Hilbert modular varieties is studied. There exists no arithmetic intersection theory for the stack H(1). Thus, following a suggestion of S. S. Kudla, we work with the tower of schemes { H(N)} N≥3 as a substitute for H(1). We define the arithmetic Chow ring for the Hilbert modular variety to be the inverse limit of the arithmetic Chow rings associated to this tower. In this way, we obtain R-valued arithmetic intersection numbers; even so, for every N, we need to calculate the arithmetic intersection numbers only up to contributions at the finite places dividing the level.
We introduce the arithmetic Hirzebruch-Zagier divisors T N (m) = (T N (m), g N (m)), where g N (m) is the pullback of the automorphic Green object described in Section 2. The subgroup of the first arithmetic Chow group generated by these divisors has finite rank (see Theorem 6.1). We let c 1 M k ( K (N)) be the first arithmetic Chern class of the pre-log singular hermitian line bundle M k ( K (N)) of modular forms of weight k with the Petersson metric.
The sequences T(m) := ( T N (m)) N≥3 and c 1 (M k ) := c 1 (M k ( K (N))) N≥3 define classes in the arithmetic Chow group CH 1 ( H, D pre ), which is defined as the inverse limit of the arithmetic Chow groups CH 1 ( H(N), D pre ).
Let M + 2 (D, χ D ) be the space of holomorphic modular forms of weight 2 for the congruence subgroup 0 (D) ⊆ SL 2 (Z) with character χ D = D · satisfying the "plus-condition" as in [HZ]. It has a natural Q-structure that is given by modular forms with rational Fourier coefficients.

THEOREM A
The arithmetic generating series To establish Theorem A, we show that the Green functions G m are compatible with the relations in the arithmetic Chow group given by Borcherds products (Theorem 4.3). Moreover, using the q-expansion principle and the results of Section 4, we prove that the divisor of a Borcherds product over Z is horizontal and therefore compatible with the definition of the Hirzebruch-Zagier divisors over Z as Zariski closures (Proposition 5.7). Now, modularity follows by a duality argument due to Borcherds in the geometric situation (see [B2]). Our main result below provides some information on the position of the generating series in the arithmetic Chow group. Its proof is much more involved since, in contrast to Theorem A, it requires the computation of arithmetic intersections.

THEOREM B
We have the following identities of arithmetic intersection numbers: Here, E(τ ) denotes the Eisenstein series defined in (4.2). In particular, the arithmetic self-intersection number of M k is given by The arithmetic intersection theory used here can be viewed as a substitute of an arithmetic intersection theory on stacks for our particular situation. Notice that any reasonable arithmetic intersection theory for the coarse moduli space without level structure has to map into the above theory so that all arithmetic degrees and heights would agree. Many arguments of the present article can (probably) be generalized to Shimura varieties of type O(2, n), where one would like to study the arithmetic intersection theory of Heegner divisors (also referred to as special divisors; see, e.g., [Ku6], [Ku1]). In fact, the work on this article already motivated generalizations of partial aspects. For instance, automorphic Green functions are investigated in [BK] and the curve lemma in [Br3]. However, progress on the whole picture will (in the near future) be limited to the exceptional cases, where regular models of such Shimura varieties are available.

Arithmetic Chow rings with pre-log-log forms
The natural metrics that appear, when considering automorphic vector bundles on noncompact locally symmetric spaces, do not extend to smooth metrics on a compactification of the space. This is the case, for instance, for the Petersson metric on the line bundle of modular forms on a Hilbert modular surface. Therefore the arithmetic intersection theory developed by Gillet and Soulé [GS1], [GS2] cannot be applied directly to study this kind of hermitian vector bundles. In the articles [BKK1] and [BKK2], there are two extensions of the arithmetic intersection theory which are suited to the study of automorphic vector bundles. The first of these theories is based on what are called pre-log and pre-log-log forms, and the second extension is based in log and log-log forms. The difference between them is that a differential form ω is pre-log or pre-log-log if ω, ∂ω,∂ω, and ∂∂ω satisfy certain growth conditions, whereas a differential form is called log or log-log form if all the derivatives of all the components of the differential form satisfy certain growth conditions. The main advantage of the second definition is that we know the cohomology computed by the log and log-log forms; therefore the arithmetic Chow groups defined with them have better properties. By contrast, the main advantage of the first approach is that it is easier to check that a particular form satisfies the conditions defining pre-log and pre-log-log forms. Since the arithmetic intersection products obtained by both theories are compatible, for simplicity we have chosen to use in this article the first of these theories.
1.1. Differential forms with growth conditions Notation 1.1 Let X be a complex algebraic manifold of dimension d, and let D be a normal crossing divisor of X. We denote by E * X the sheaf of smooth complex differential forms on X. Moreover, we write U = X \ D, and we let j : U → X be the inclusion.
Let V be an open coordinate subset of X with coordinates z 1 , . . . , z d ; we put r i = |z i |. We say that V is adapted to D if the divisor D is locally given by the equation z 1 · · · z k = 0. We assume that the coordinate neighborhood V is small enough; more precisely, we assume that all the coordinates satisfy r i < 1/e e , which implies that log 1/r i > e and log(log 1/r i ) > 1.
If f and g are two complex functions, we write f ≺ g if there exists a constant C > 0 such that |f (x)| ≤ C|g(x)| for all x in the domain of definition under consideration.

Definition 1.2
We say that a smooth complex function f on X \ D has log-log growth along D if we have for any coordinate subset V adapted to D and some positive integer M. The sheaf of differential forms on X with log-log growth along D is the subalgebra of j * E * U generated, in each coordinate neighborhood V adapted to D, by the functions with log-log growth along D and the differentials A differential form with log-log growth along D is called a log-log growth form.

Definition 1.3
A log-log growth form ω such that ∂ω,∂ω, and ∂∂ω are also log-log growth forms is called a pre-log-log form. The sheaf of pre-log-log forms is the subalgebra of j * E * U generated by the pre-log-log forms. We denote this complex by E * X D pre .
In [BKK2, Proposition 7.6], it is shown that pre-log-log forms are integrable and the currents associated to them do not have residues.
The sheaf E * X D pre , together with its real structure, its bigrading, and the usual differential operators ∂,∂, is easily checked to be a sheaf of Dolbeault algebras. We call it the Dolbeault algebra of pre-log-log forms. Observe that it is the maximal subsheaf of Dolbeault algebras of the sheaf of differential forms with log-log growth.

Definition 1.4
We say that a smooth complex function f on U has log growth along D if we have for any coordinate subset V adapted to D and some positive integer M. The sheaf of differential forms on X with log growth along D is the subalgebra of j * E * U generated, in each coordinate neighborhood V adapted to D, by the functions with log growth along D and the differentials A differential form with log growth along D is called a log growth form.
Definition 1.5 A log growth form ω such that ∂ω,∂ω, and ∂∂ω are also log growth forms is called a pre-log form. The sheaf of pre-log forms is the subalgebra of j * E * U generated by the pre-log forms. We denote this complex by E * X D pre .
The sheaf E * X D pre , together with its real structure, its bigrading, and the usual differential operators ∂,∂, is easily checked to be a sheaf of Dolbeault algebras. We call it the Dolbeault algebra of pre-log forms. It is the maximal subsheaf of Dolbeault algebras of the sheaf of differential forms with log growth.
For the general situation of interest to us, we need a combination of the concepts of pre-log-log and pre-log forms. Notation 1.6 Let X, D, U , and j be as above. Let D 1 and D 2 be normal crossing divisors, which may have common components, and such that D = D 1 ∪ D 2 . We denote by D 2 the union of the components of D 2 which are not contained in D 1 . We say that the open coordinate subset V is adapted to D 1 and D 2 , if D 1 has the equation z 1 · · · z k = 0, D 2 has the equation z k+1 · · · z l = 0, and r i = |z i | < 1/e e for i = 1, . . . , d.

Definition 1.7
We define the sheaf of differential forms with log growth along D 1 and log-log growth along D 2 to be the subalgebra of j * E * U generated by differential forms with log growth along D 1 and log-log growth along D 2 .
A differential form with log growth along D 1 and log-log growth along D 2 is called a mixed growth form if the divisors D 1 and D 2 are clear from the context.

Definition 1.8
Let X, D = D 1 ∪ D 2 , U , and j be as before. A mixed growth form ω such that ∂ω,∂ω, and ∂∂ω are also mixed growth forms is called a mixed form. The sheaf of mixed forms is the subalgebra of j * E * U generated by the mixed forms. We denote this The sheaf E * X D 1 D 2 pre , together with its real structure, its bigrading, and the usual differential operators ∂,∂, is easily checked to be a sheaf of Dolbeault algebras. We call it the Dolbeault algebra of mixed forms. Observe that we have, by definition, 1.2. Pre-log-log Green objects Notation 1.9 Let X be a complex algebraic manifold of dimension d, and let D be a normal crossing divisor. We denote by X the pair (X, D).
In the sequel, we consider all operations adapted to the pair X. For instance, if Y X is a closed algebraic subset and W = X \ Y , then an embedded resolution of singularities of Y in X is a proper modification π : X → X such that π| π −1 (W ) : π −1 (W ) → W is an isomorphism, and are normal crossing divisors on X. Using Hironaka's theorem on the resolution of singularities (see [Hi]), one can see that such an embedded resolution of singularities exists.
Analogously, a normal crossing compactification of X is a smooth compactification X such that the closure D of D, the subset B X = X \ X, and the subset B X ∪ D are normal crossing divisors.
Given a diagram of normal crossing compactifications of X , with divisors B X and B X at infinity, respectively, then by functoriality of mixed forms, there is an induced morphism In order to have a complex that is independent of the choice of a particular compactification, we take the limit over all possible compactifications. Namely, we denote where the limit is taken over all normal crossing compactifications X of X. The assignment that sends an open subset U of X to E * pre (U ) is a totally acyclic sheaf in the Zariski topology (see [BKK1,Remark 3.8 ]); we denote it by E * pre,X .
Definition 1.10 Let X = (X, D) be as above. Then we define the complex E * pre (X) of differential forms on X, pre-log along infinity, and pre-log-log along D as the complex of global sections of E * pre,X ; that is, Let X be a smooth real variety, and let D be a normal crossing divisor defined over R; as before, we write X = (X, D). For any U ⊆ X, the complex E * pre (U ) is a Dolbeault algebra with respect to the wedge product. For any Zariski open subset U ⊆ X, we put where D * (E pre (U C ), * ) is the Deligne algebra associated to the Dolbeault algebra E * pre (U ) and σ is the antilinear involution ω → F ∞ (ω) (see [BKK2,Definition 7.18]). When X = (X, D) is clear from the context, we write D * pre (U, * ) instead of D * pre,X (U, * ).
The arithmetic complex D pre made out of pre-log and pre-log-log forms can be seen as the complex that satisfies the minimal requirement needed to allow log-log singularities along a fixed divisor with normal crossing as well as to have a theory of arithmetic intersection numbers. Observe that such singularities naturally occur if one works with automorphic vector bundles (see [M]). Let U → X be an open immersion, and let Y = X \ U . For integers n, p, we write where the latter groups are truncated relative cohomology groups (see [BKK2, Definition 2.55]). Recall that a class g ∈ H n D pre ,Y (X, p) is represented by a pair g = (ω, g), with ω ∈ Z(D n pre (X, p)) a cocycle and g ∈ D n−1 pre (U, p) := D n−1 pre (U, p) Im d D pre , such that d D pre g = ω. There are morphisms given by ω(g) = ω(ω, g) = ω, and surjective morphisms given by sending the class of the pair (ω, g) in H n D pre ,Y (X, p) to its class [ω, g] in the cohomology group H n (D * pre (X, p), D * pre (U, p)).

Definition 1.12
Let y be a p-codimensional algebraic cycle on X with supp y ⊆ Y . A weak pre-loglog Green object for y (with support in Y ) is an element here, the class cl(y) is given by the image of the class of the cycle y in real Deligne-Beilinson cohomology via the natural morphism H 2p D,Y (X, R(p)) → H 2p D pre ,Y (X, p). If Y = supp y, then g y is called a pre-log-log Green object for y.
The surjectivity of the morphism cl implies that any algebraic cycle as before has a weak pre-log-log Green object with support in Y . For the convenience of the reader, we now recall that where d = ∂ +∂ and ∂, respectively,∂, are the usual holomorphic, respectively, antiholomorphic, derivatives. Then a weak pre-log-log Green object for y, as above, is represented by a pair Observe that a weak pre-log-log Green object carries less information than a prelog-log Green object. For instance, the subsequent proposition is not true, in general, for weak Green objects. Assume that y = j n j Y j with irreducible subvarieties Y j and certain multiplicities n j . If the cycle (ω, g) represents the class of y, then the equality holds for any differential form α; here, Notice that in contrast to the theory by Gillet and Soulé, a pre-log-log Green object is not characterized by (1.3). However, if ω is a smooth form, then a pre-log-log Green object determines a Green current in the sense of Gillet and Soulé.
In the sequel, we use (for a cycle y, as above) the shorthand where, as usual,

Star products of pre-log-log Green objects
Let X = (X, D) be a proper smooth real variety of dimension d with fixed normal crossing divisor D. Moreover, let Y, Z be closed subsets of X. Then it is shown in [BKK2] that the product • of Deligne-Beilinson cohomology induces a star product * : which is graded commutative, associative, and compatible with the morphisms ω and cl. We fix a cycle y ∈ Z p (X R ) with supp y ⊆ Y and a cycle z ∈ Z q (X R ) with supp z ⊆ Z. If y and z intersect properly, there is a well-defined intersection cycle y · z. If they do not intersect properly, then the cycle y · z is defined in the Chow group of X with supports on Y ∩ Z. Let g y = (ω y , g y ) ∈ H 2p D pre ,Y (X, p) and g z = (ω z , g z ) ∈ H 2q D pre ,Z (X, q) be weak pre-log-log Green objects for y and z, respectively. Then g y * g z is a weak pre-log-log Green object for the cycle y · z with support Y ∩ Z. Moreover, if supp y = Y , supp z = Z, and Y and Z intersect properly, then g y * g z is a pre-log-log Green object for the cycle y · z. We now recall how to find a representative of this Green object.
Adapting the argument of [Bu,p. 362], we can find an embedded resolution of singularities of Y ∪ Z, π : X R → X R , which factors through embedded resolutions of Y , Z, and Y ∩ Z. In particular, we can assume that are also normal crossing divisors. Let us denote by Y the normal crossing divisor formed by the components of π −1 (Y ) which are not contained in π −1 (Y ∩ Z). Analogously, we denote by Z the normal crossing divisor formed by the components of π −1 (Z) which are not contained in π −1 (Y ∩ Z). Then Y and Z are closed subsets of X which do not meet. Therefore there exist two smooth, F ∞ -invariant functions σ Y Z and σ ZY satisfying 0 ≤ σ Y Z , σ ZY ≤ 1, σ Y Z + σ ZY = 1, σ Y Z = 1 in a neighborhood of Y , and σ ZY = 1 in a neighborhood of Z. Finally, in the group H 2p+2q D pre ,Y ∩Z (X, p + q), we then have the identity (1.5) In order to compute the arithmetic degree of an arithmetic intersection, we need formulas for the pushforward of certain * -products in the top degree of truncated cohomology groups. We make the convention that for g = (ω, g) ∈ H 2d+2 D pre ,∅ (X, d), we write X g instead of X g. THEOREM 1.14 Let X be as before, and assume that D = D 1 ∪ D 2 , where D 1 and D 2 are normal crossing divisors of X satisfying D 1 ∩ D 2 = ∅. Let y and z be cycles of X such that supp y = Y and supp z ⊆ Z. Let g y be a pre-log-log Green object for y, and let g z be a weak pre-log-log Green object for z with support Z. Assume that p + q = d + 1, and assume that Y ∩ Z = ∅, Y ∩ D 2 = ∅, and Z ∩ D 1 = ∅. Then where B ε (D j ) denotes an ε-neighborhood of D j (j = 1, 2) and B ε (D) = B ε (D 1 ) ∪ B ε (D 2 ).

Proof
Let X be an embedded resolution of Y ∪ Z as described above. We write Y , respectively, Z , for the strict transforms of Y , respectively, Z; we note that Y ∩ Z = ∅. Furthermore, we write Y for the strict transform of the closure of Y \(Y ∩D). Choosing σ Y Z and σ ZY as above, we may assume that σ Y Z has value 1 in a neighborhood of D 1 (since D 1 ∩ Z = ∅) and vanishes in a neighborhood of D 2 .
Using −2∂∂ = (4πi) dd c , we get, by means of (1.5), (1.6) In order to perform the following calculations, we put where B ε (·) denotes an ε-neighborhood of the quantities in question. On X ε , one can split up the integral in question by means of [BKK2, (7.32)]: (1.7) Applying Stokes's theorem to the latter integral and using the properties of the function σ Y Z , we obtain, for sufficiently small ε > 0, Taking into account that (ω y , g y ) is a Green object for y and that g z is a smooth (1.8) Here, f (ε) is a continuous function with lim ε→0 f (ε) = 0. Combining (1.6), (1.7), and (1.8), we finally find Hence the claim follows.

ᮀ
Observe that if D is empty and supp z = Z, then the formula of Theorem 1.14 specializes to the formula for the star product given by Gillet and Soulé (see [GS1]). Nevertheless, when D is not empty, both terms in the formula of Gillet and Soulé may be divergent. Therefore one can view the theory of cohomological arithmetic Chow groups as a device that gives, in the nonsmooth situation, the necessary correction terms.

Arithmetic Chow rings with pre-log-log forms
Let K be a number field, let A be a subring of K with field of fractions K, and let be a complete set of complex embeddings of K into C. Let X be an arithmetic variety over A of (relative) dimension d over S = Spec A (i.e., a regular scheme X that is flat and quasi-projective over Spec A). We let X ∞ = σ ∈ X σ (C). This complex variety has a natural antilinear involution denoted F ∞ . We denote by X R the real variety defined by X ∞ and F ∞ . Let D K be a fixed normal crossing divisor of X K . We denote by D the induced normal crossing divisor on X R . In this section, we recall basic properties of the arithmetic Chow groups CH Let Z p (X ) be the group formed by cycles on X of codimension p. Given y ∈ Z p (X ), we write y ∞ = σ ∈ y σ (C) and let Y = supp y ∞ . We define where in the limit, Z p is the set of cycles on X ∞ of codimension at least p ordered by inclusion.

Definition 1.15
The group of p-codimensional arithmetic cycles on X is the group Let w be a codimension p − 1 irreducible subvariety of X , and let h ∈ k(w) * . Write h ∞ for the induced function on w ∞ , and set Y = supp(div(h ∞ )). Then there is a distinguished pre-log-log Green object g(h) ∈ H 2p D pre ,Y (X , p) for div(h ∞ ). We point out that g(h) depends only on the class of h ∞ in H 2p−1 D pre (X \ Y, p). We write div(h) = (div(h), g(h)) for this arithmetic cycle, and we denote by Rat p (X , D pre ) the subgroup of Z p (X , D pre ) generated by arithmetic cycles of the form div(h). Then the pth arithmetic Chow group of X with log-log growth along D is defined by A key result of [BKK2] is the definition of an arithmetic intersection product equipped with this product has the structure of a commutative associative ring. We call CH * (X , D pre ) Q the arithmetic Chow ring of X with log-log growth along D (for a detailed description of the arithmetic intersection product, we refer to [BKK2,Theorems 4.18,4.19]). We now briefly discuss the special case of p + q = d + 1. Let (y, g y ) ∈ Z p (X , D pre ) and (z, g z ) ∈ Z q (X , D pre ) be such that y ∞ and z ∞ have proper intersection on X . Since p + q = d + 1, this means y ∞ ∩ z ∞ = ∅, and the intersection of y and z defines a class [y · z] fin in the Chow group with finite support CH d+1 fin (X ) Q . One obtains (1.9) Definition 1.16 Let K be a number field, let O K be its ring of integers, and let be a complete set of complex embeddings of K into C. Then Spec O K is an arithmetic variety, and due to the product formula for K, we have as in [SABK] a well-defined arithmetic degree map (1.10) induced by the assignment In particular, this map is a group homomorphism, which is an isomorphism in the case of K = Q; it is common to identify CH 1 (Spec(Z), D pre ) with R. If X is a d-dimensional projective arithmetic variety over A and π : X → S is the structure morphism, another key result of [BKK2] is the definition of a pushforward morphism π * : CH d+1 (X , D pre ) −→ CH 1 (S, D pre ).
The arithmetic Chow ring CH * (X , D pre ) is a generalization of the classical construction CH * (X ) due to Gillet and Soulé (see, e.g., [SABK]), in which the differential forms are allowed to have certain log-log singularities along a fixed normal crossing divisor.

THEOREM 1.18
If X is projective, then there is a commutative diagram in which the upper morphism is compatible with the product structure, the vertical morphisms are the pushforward morphisms, and the lower morphism is an isomorphism compatible with arithmetic degrees. In particular, this diagram implies the compatibility of the arithmetic intersection numbers that can be computed in both theories.

Remark 1.19
Let A be as in Remark 1.17; then the morphism Here, we used the convention that for g P = (ω P , g P ), we write X ∞ g P instead of In order to ease notation, we sometimes write for α ∈ CH d+1 (X , D pre ) simply α instead of deg π * (α).

Pre-log singular hermitian line bundles and Faltings heights
Let X and D be as in Section 1.4.

Definition 1.20
Let L be a line bundle on X equipped with an F ∞ -invariant singular hermitian metric · on the induced line bundle L ∞ over X ∞ . If there is an analytic trivializing cover {U α , s α } α such that, for all α, then the metric is called a pre-log singular hermitian metric. The pair (L , · ) is called a pre-log singular hermitian line bundle and denoted by L .

LEMMA 1.21
If L is a pre-log singular hermitian line bundle on X , then for any rational section (1.12)

Proof
Let · 0 be an F ∞ -invariant smooth hermitian metric on the line bundle L ∞ . Since the quotient s / s 0 does not depend on the section s, Definition 1.20 implies that is a pre-log-log function. Consequently, (2∂∂f, −f ) is a pre-log-log Green object for the empty divisor (see [BKK2, Section 7.7]). Since (2∂∂ log s 0 , − log s 0 ) is a Green object for div(s) and we obtain that (2∂∂ log s , − log s ) is a pre-log-log Green object for div(s).

ᮀ
It is easy to see that the class of (1.12) depends only on the pair (L , · ). We denote it by c 1 (L ), and we call it the first arithmetic Chern class of L .

Definition 1.22
The arithmetic Picard group Pic(X , D pre ) is the group of isomorphy classes of prelog singular hermitian line bundles, where the group structure is given by the tensor product.
We have an inclusion Pic(X ) ⊆ Pic(X , D pre ), where Pic(X ) is the arithmetic Picard group defined by Gillet and Soulé. Moreover, the morphism given by equation (1.12), is an isomorphism. Finally, given a pre-log singular hermitian line bundle L on an arithmetic variety of (relative) dimension d over A, we write and we call it the arithmetic self-intersection number of L .
is defined. Observe that since the height of a cycle, whose generic part is supported in D, may be infinite, one cannot expect that the height pairing due to Bost, Gillet, and Soulé [BGS] unconditionally generalizes to a height pairing between the arithmetic Chow groups CH p (X , D pre ) and the whole group of cycles Z q (X ).
We now let X , D, U be as before, and we let p, q be integers satisfying p + q = d + 1. Let z ∈ Z q U (X ) be an irreducible, reduced cycle, and let α ∈ CH p (X , D pre ).
We represent α by the class of an arithmetic cycle (y, g y ), where y is a p-codimensional cycle such that y K intersects z K properly and where g y = (ω y , g y ) is a pre-log-log Green object for y. We have Here, the quantity π # (g y ∧ δ z ) has to be understood as follows. Let Z = supp z R , and let ı : Z → Z be a resolution of singularities of Z adapted to D. Since y K ∩ z K = ∅, the functoriality of pre-log-log forms shows that ı * (g y ) is a pre-log-log form on Z; hence it is locally integrable on Z, and we have The pairing (1.14) is now obtained by linearly extending the above definitions.
If we choose a basic pre-log-log Green form g z for z and put g z = (−2∂∂g z , g z ), then the height pairing (1.14) satisfies The height pairing (1.14) is of particular interest when α = c 1 (L ) p for some pre-log singular hermitian line bundle L on X . We call the real number the Faltings height of z (with respect to L ).

Remark 1.23
As in Theorem 1.18, if the hermitian metric of L is smooth, the Faltings height computed in this arithmetic intersection theory agrees with the one defined by Bost, Gillet, and Soulé.

Complex theory of Hilbert modular surfaces
We begin by recalling some basic facts on Hilbert modular surfaces. This mainly serves to fix notation (for a detailed account, we refer to [Fr3] and [G]). Let K be a real quadratic field with discriminant D. Let O K be its ring of integers, and let d = ( √ D) be the different. We write x → x for the conjugation in K, tr(x) = x + x for the trace, and N(x) = xx for the norm of an element. Given an a ∈ K, we write a 0 if a is totally positive. Furthermore, we denote by ε 0 > 1 the fundamental unit of K. We write χ D for the quadratic character associated with K given by the Legendre symbol χ D (x) = D x . The Dirichlet L-function corresponding to χ D is denoted by L(s, χ D ). Moreover, we write ζ (s) for the Riemann zeta function.
Let H = {z ∈ C; (z) > 0} be the upper complex half-plane. The group SL 2 (R) × SL 2 (R) acts on the product H 2 of two copies of H via Möbius transformations on both factors. As usual, we identify SL 2 (K) with a subgroup of for the Hilbert modular group corresponding to a. Moreover, we briefly write and we denote by K (N) the principal congruence subgroup of level N, that is, the kernel of the natural homomorphism K → SL 2 (O K /NO K ). Throughout, we use z = (z 1 , z 2 ) as a standard variable on H 2 . We denote its real part by (x 1 , x 2 ) and its imaginary part by (y 1 , y 2 ). Let ≤ SL 2 (K) be a subgroup that is commensurable with K . The quotient \H 2 is called the Hilbert modular surface associated with . It is a noncompact normal complex space that can be compactified by adding the cusps of , that is, the -classes of P 1 (K). By the theory of Baily and Borel, the quotient together with the Baily-Borel topology, can be given the structure of a normal projective algebraic variety over C. It is called the Baily-Borel compactification of \H 2 . Recall that the cusps of (O K ⊕ a) are in bijection with the ideal classes of K by mapping (α : β) ∈ P 1 (K) to the ideal αO K + βa −1 . So, in particular, the cusp ∞ corresponds to the principal class and 0 to the class of a −1 . For any point ξ ∈ H 2 ∪P 1 (K), we denote by ξ the stabilizer of ξ in . If ξ ∈ H 2 , then the quotient G = ξ /{±1} is a finite cyclic group. If |G| > 1, then ξ is called an elliptic fixed point. Notice that K always has elliptic fixed points of orders 2 and 3. On the other hand, K If a is a fractional ideal from the principal genus of K, there are a fractional ideal c and a totally positive λ ∈ K such that a = λc 2 . If M denotes a matrix in ( 2.4) This induces an isomorphism of algebraic varieties over C, , and c −1 is mapped to the cusp 0.

Desingularization and the curve lemma
Throughout, we write e(z) = e 2πiz . We denote by E(δ) = {q ∈ C; |q| < δ} the δ-disc around the origin, and we put E = E(1). Moreover, we writeĖ = {q ∈ E; q = 0}. The singular locus X( ) sing of X( ) consists of the cusps and the elliptic fixed points. Throughout, we work with desingularizations of X( ) such that the pullback of the singular locus is a divisor with normal crossings. Given such a desingularization (2.6) we denote this divisor by We now present a local description of X( ) using the "curve lemma" due to Freitag. Let κ ∈ P 1 (K) be a cusp of , and let g ∈ SL 2 (K) with κ = g∞. By replacing by the commensurable group g −1 g, we may assume that κ = ∞. There are a Z-module t ⊂ K of rank 2 and a finite index subgroup of the units of O K acting on t such that ∞ has finite index in the semidirect product t (see [G,Chapter 2.1]). In particular, if γ ∈ ∞ , then γ (z 1 , z 2 ) = (εz 1 + µ, ε z 2 + µ ) for some µ ∈ t and some totally positive unit ε ∈ . A fundamental system of open neighborhoods of ∞ ∈ X( ) is given by (2.8) Let C > 0. Let a ∈ X( ) be a point with π(a) = ∞, and let U ⊂ X( ) be a small open neighborhood of a such that π(U ) ⊂ V C . Possibly replacing U by a smaller neighborhood, after a biholomorphic change of coordinates we may assume that U = E 2 is the product of two unit discs, a = (0, 0), and that π * ∞ = div(q α 1 q β 2 ) on U with nonnegative integers α, β.

Remark 2.2
The properties of the Baily-Borel topology on X( ) imply that the exceptional divisor π * (∞) contains the component {q j = 0} if and only if µ j is totally positive. Now, let ξ ∈ H 2 be an elliptic fixed point of , let G be the cyclic group ξ /{±1}, and let n = |G|. Let V ⊂ H 2 be a small open neighborhood of ξ on which ξ acts. Then ξ \V is an open neighborhood of ξ ∈ \H 2 . Let a ∈ X( ) be a point with π(a) = ξ , and let U ⊂ X( ) be an open neighborhood of a such that π(U ) ⊂ ξ \V . Without loss of generality, we may assume that U = E 2 is the product of two unit discs, a = (0, 0), and that π * ξ = div(q α 1 q β 2 ) on U . The desingularization map induces a holomorphic map E 2 → ξ \V . Arguing as in [Fr2,Hilfssatz 5.19,p. 200] and using the fact that |G| = n, we get a commutative diagram of holomorphic maps (2.13) From this, one derives an analogue of the curve lemma for the elliptic fixed points.
There is a (unique up-to-a-positive multiple) symmetric (SL 2 (R) × SL 2 (R))invariant Kähler metric on H 2 . Its corresponding (1, 1)-form is given by (2.14) The form ω induces a pre-log-log form on X( ) with respect to D .

Proof
We show that if κ is a cusp of , and a ∈ X( ) with π(a) = κ, then π * ω satisfies the growth conditions of Definition 1.2 in a small neighborhood of a. The corresponding assertion for the elliptic fixed points is easy and is left to the reader. Without loss of generality, we may assume that κ = ∞ and that π looks locally near a as in (2.9).
By means of the ∞ -invariant function log(y 1 y 2 ), we may write ω = (1/(2πi))∂∂ log(y 1 y 2 ). Using the notation of Lemma 2.1, we see that * y 1 y 2 = − 1 2π Consequently, Hence we find Since H is holomorphic, this differential form has log-log growth along π * (∞). ᮀ It follows that the volume form is also a pre-log-log form. It is well known that (see, e.g., [G, p. 59]). Here, ζ K (s) is the Dedekind zeta function of K.

Hilbert modular forms and the Petersson metric
Let k be an integer, and let χ be a character of . A meromorphic function F on H 2 is called a Hilbert modular form of weight k (with respect to and χ) if it satisfies If F is holomorphic on H 2 , it is called a holomorphic Hilbert modular form. Then, by the Koecher principle, F is automatically holomorphic at the cusps. We denote the vector space of holomorphic Hilbert modular forms of weight k (with respect to and trivial character) by M k ( ). A holomorphic Hilbert modular form F has a Fourier expansion at the cusp ∞ of the form ( 2.21) and it has analogous expansions at the other cusps. The sum runs through all totally then t is equal to ad −1 . Any Hilbert modular form is the quotient of two holomorphic forms. We say that a Hilbert modular form has rational Fourier coefficients, if it is the quotient of two holomorphic Hilbert modular forms with rational Fourier coefficients. Meromorphic (holomorphic) modular forms of weight k can be interpreted as rational (global) sections of the sheaf M k ( ) of modular forms. If we write p : H 2 → \H 2 for the canonical projection, then the sections over an open subset U ⊂ \H 2 are holomorphic functions on p −1 (U ), which satisfy the transformation law (2.20). This defines a coherent analytic sheaf on \H 2 , which is actually algebraic. By the Koecher principle, it extends to an algebraic sheaf on X( ). By the theory of Baily and Borel, there is a positive integer n such that M k ( ) is a line bundle if n |k, and [G, p. 44], [C, p. 549]). The line bundle of modular forms of weight k (divisible by n ) on X( ) is defined as the pullback π * M k ( ). By abuse of notation, we also denote it by M k ( ). In the same way, if F is a Hilbert modular form of weight k, we simply write F for the section π * (F ) on X. The divisor div (F ) Here, div(F ) denotes the strict transform of the divisor of the modular form F on X( ), and E j are the irreducible components of the exceptional divisor D . The multiplicities n j are determined by the orthogonality relations div(F ) · E j = 0.

Definition 2.4
If F ∈ M k ( )(U ) is a rational section over an open subset U ⊂ \H 2 , we define its Petersson metric by This defines a hermitian metric on the line bundle of modular forms of weight k on \H 2 . We now study how it extends to X( ).

PROPOSITION 2.5
The Petersson metric on the line bundle M k ( ) of modular forms on X( ) is a pre-log singular hermitian metric (with respect to D ).

Proof
We have to verify the conditions of Definition 1.20 locally for the points of D . Here, we consider only the points above the cusps of X( ). The corresponding assertion for the elliptic fixed points (if there are any) is left to the reader. Let κ be a cusp, and let a ∈ X( ) with π(a) = κ. Moreover, let F be a trivializing section of M k ( ) over a small neighborhood of a. We have to show that log F Pet satisfies the growth conditions of Definition 1.3. Without loss of generality, we may assume that κ = ∞ and that π looks locally near a as in (2.9). It suffices to show that π * log(y 1 y 2 ) is a pre-log-log form near a.
That π * log(y 1 y 2 ) and ∂∂π * log(y 1 y 2 ) have log-log growth along D follows from (2.16) and (2.17) in the proof of Lemma 2.3. Using the notation of Lemma 2.3, we see that∂ Since H is holomorphic, we may infer that∂π * log(y 1 y 2 ) has log-log growth along D . Analogously, we see that ∂π * log(y 1 y 2 ) has log-log growth.

ᮀ
The first Chern form c 1 (M k ( ), · Pet ) of the line bundle M k ( ) equipped with the Petersson metric is given by (2.22)

Definition 2.6
If F is a Hilbert modular form for , then we denote the Green object for div (F ) by (2.23)

Remark 2.7
In view of (2.19), the geometric self-intersection number M k ( ) 2 of the line bundle of modular forms of weight k is equal to k 2 [ K : ] ζ K (−1).

Green functions for Hirzebruch-Zagier divisors
From now on, we assume that the discriminant D of the real quadratic field K is a prime. This implies that the fundamental unit ε 0 has norm −1.
We consider the rational quadratic space V of signature (2, 2) of matrices A = a ν ν b with a, b ∈ Q and ν ∈ K, with the quadratic form q(A) = det(A). For a fractional ideal a of K, we consider the lattices ( 2.25) Notice that the dual of L(a) is (1/N(a))L (a). The group SL 2 (K) acts on V by γ.A = γ Aγ t for γ ∈ SL 2 (K). Under this action, (O K ⊕a −1 ) preserves the lattices L(a) and L (a). In particular, one obtains an injective homomorphism ( Let m be a positive integer. Recall that the subset It is the inverse image of an algebraic divisor on the quotient (O K ⊕ a)\H 2 , which is also denoted by T a (m). Here, we understand that all irreducible components of T a (m) are assigned the multiplicity 1. (There is no ramification in codimension 1.) The divisor T a (m) is nonzero if and only if χ D (m) = −1. If m is square free, then since D is prime, T a (m) is irreducible (see [HZ], [G,Chapter 5]). Moreover, T a (m) and T a (n) intersect properly if and only if mn is not a square.
Since there is only one genus, there are a fractional ideal c and a totally positive This implies that the isomorphism (2.5) takes We are mainly interested in the case of a = O K . To lighten the notation, we briefly write L = L (O K

Definition 2.8
Let m be a positive integer with χ D (m) = −1. If m is the norm of an ideal in O K , then T (m) is a noncompact divisor on K \H 2 , birational to a linear combination of modular curves. In this case, we say that T (m) is isotropic. If m is not the norm of an ideal in O K , then T (m) is a compact divisor on K \H 2 , birational to a linear combination of Shimura curves. In that case, we say that T (m) is anisotropic.
These notions are compatible with the description of K \H 2 and the divisors T (m) as arithmetic quotients corresponding to orthogonal groups of type O(2, 2) and O(2, 1), respectively (see Section 4.2). Here, isotropic (anisotropic) Hirzebruch-Zagier divisors are given by isotropic (anisotropic) rational quadratic spaces of signature (2, 1).
In [Br1, Section 3], a certain Green function m (z 1 , z 2 , s) was constructed which is associated to the divisor T (m). We briefly recall some of its properties. For s ∈ C with (s) > 1, the function m (z 1 , z 2 , s) is defined by Here, Q s−1 (t) is the Legendre function of the second kind (see [AS,Section 8]), defined by The sum in (2.27) converges normally for (s) > 1 and (z 1 , z 2 ) ∈ H 2 − T (m). This implies that m (z 1 , z 2 , s) is invariant under K . It has a Fourier expansion which converges for y 1 y 2 > m/D and (z 1 , z 2 ) / ∈ T (m). As a function in s, the latter sum over ν = 0 converges normally for (s) > 3/4. The constant term is a meromorphic function in s with a simple pole at s = 1. A refinement of these facts can be used to show that m (z 1 , z 2 , s) has a meromorphic continuation in s to {s ∈ C; (s) > 3/4}. Up to a simple pole in s = 1, it is holomorphic in this domain (see [Br1,Theorem 1]).
We denote by The Fourier expansion (2.29) of m (z 1 , z 2 , s) was determined in [Br1]. It follows from [Br1,identity (19), Lemmas 1, 2] that the constant term is given by (2.31) Notice that our G a (m, ν) equals G a (−m, ν) in the notation of [Z1]. For the purposes of the present article, we need to compute u 0 (y 1 , y 2 , s) more explicitly. We define a generalized divisor sum of m by (2.32) It satisfies the functional equation σ m (s) = σ m (−s). If p is a prime and n ∈ Z, then we denote by v p (n) the additive p-adic valuation of n.
If m is square free and coprime to D, then The first formula follows from the multiplicativity of σ m (s). The second can be obtained from the Euler product expansion in a straightforward way.

LEMMA 2.10
We have

Proof
The exponential sum G a (m, 0) is equal to as in [Z1, p. 27]. It is easily seen that N b (n) is multiplicative in b. Hence it suffices to determine N b (n) for prime powers b = p r . We get the following Euler product expansion: The function N p r (n) can be determined explicitly by means of [Z1, Lemma 3]. By a straightforward computation, we find that the local Euler factors are equal to for p prime with (p, D) = 1. Inserting this into (2.34), we obtain the assertion by means of Lemma 2.9.
ᮀ Hence the second term in (2.30) is equal to By virtue of the functional equation we may rewrite (2.35) in the form Using the Legendre duplication formula, (s − 1/2) (s) = √ π2 2−2s (2s − 1), we finally obtain, for the second term in (2.30), In the following, we compute the first summand in (2.30). The subset of H 2 is a union of hyperplanes of real codimension 1. It is invariant under the stabilizer of the cusp ∞. Following the notation of [B1], we call the connected components of H 2 − S(m) the Weyl chambers of discriminant m. For a subset W ⊂ H 2 and λ ∈ K, we write (λ, W ) > 0 if λy 1 + λ y 2 > 0 for all (z 1 , z 2 ) ∈ W .
Let W ⊂ H 2 be a fixed Weyl chamber of discriminant m, and let W ⊂ W be a nonempty subset. There are only finitely many λ ∈ d −1 such that λ > 0, N(λ) = −m/D, and Denote the set of these λ by R (W , m). It is easily seen that R(W , m) = R (W, m) for all nonempty subsets W ⊂ W . By Dirichlet's unit theorem, the set of all λ ∈ d −1 with N(λ) = −m/D is given by ± λε 2n 0 ; λ ∈ R (W, m), n ∈ Z .

THEOREM 2.11
Let W ⊂ H 2 be a Weyl chamber of discriminant m. For (z 1 , z 2 ) ∈ W , the constant term of the Fourier expansion of m (z 1 , z 2 , s) is given by λ∈R (W,m) ε 2−4s 0 (λy 1 ) 1−s (−λ y 2 ) s As a function in s, the Green function m (z 1 , z 2 , s) has a simple pole at s = 1 coming from the factor ζ (2s − 1) in the first term of u 0 (y 1 , y 2 , s). However, it can be regularized at this place by defining m (z 1 , z 2 ) to be the constant term of the Laurent expansion of m (z 1 , z 2 , s) at s = 1 (see [Br1,p. 66]). Using the Laurent expansion ζ (2s − 1) (2.41) By means of the Laurent expansion of (s), one infers that L m is more explicitly given by (2.42) In later applications, it is convenient to write the regularized function m (z 1 , z 2 ) as a limit. In view of (2.40), we find that (2.43) The Fourier expansion of m (z 1 , z 2 ) can be deduced from (2.29) by virtue of Theorem 2.11 and (2.40). It is given by On a Weyl chamber W of discriminant m, we get (W,m) λ ( 2.45) is the so-called Weyl vector associated with W and m.
In order to get a Green function with a "good" arithmetic normalization, which is compatible with our normalization of the Petersson metric, we have to renormalize as follows.

Definition 2.13
We define the normalized Green function for the divisor T (m) by According to (2.44) and [Br1,Section 3.3], we have (2.46) Here, W is a Weyl chamber of discriminant m and ρ W the corresponding Weyl vector. Moreover, o(z 1 , z 2 ) is a K,∞ -invariant function on H 2 , which defines a smooth function in the neighborhood V m/D of ∞ and vanishes at ∞. This describes the singularities of G m (z 1 , z 2 ) near the cusp ∞. Analogous expansions hold at the other cusps.
We now consider G m (z 1 , z 2 ) as a singular function on X( ). For this purpose, we also write T (m) for the closure of T (m) ⊂ \ H 2 in the Baily-Borel compactification X( ).

LEMMA 2.14
The divisor T (m) on X( ) is a Q-Cartier divisor.

Proof
We show that there exists an integer n such that nT (m) is given locally as the divisor of a holomorphic function. This is clear for the restriction of T (m) to \H 2 because there are only finitely many singular points that are finite quotient singularities. Hence it suffices to consider T (m) locally at the cusps where trivializing holomorphic functions can be constructed explicitly using local Borcherds products (see [BF]). For instance, for the cusp ∞, the function is a local Borcherds product in the sense of [BF]. There exists a positive integer n such that nρ W ∈ d −1 . (By (2.45), one can take n = tr(ε 0 ).) Then ∞ m (z 1 , z 2 ) n defines a holomorphic function in a small neighborhood of ∞, whose divisor equals the restriction of nT (m).

Remark 2.15
This lemma allows us to define the pullback π * T (m) as a Q-Cartier divisor on X( ).

Remark 2.16
Observe that π * T (m) may contain components of the exceptional divisor D . This is actually always the case if T (m) is isotropic. If X( K ) is the Hirzebruch desingularization of X( K ), then π * T (m) is equal to the divisor T c m considered by Hirzebruch and Zagier in [HZ].
defines a pre-log-log Green object for the Q-divisor π * (T (m)) on X( ).

Proof
Let n be a positive integer such that nT (m) is a Cartier divisor. We define a metric on the line bundle O π * (nT (m)) on X( ) by giving the canonical section 1 O(π * (nT (m))) the norm By [Br1,Section 3.3] and formula (38), this metric is smooth outside D . We now show that it is a pre-log singular hermitian metric in the sense of Definition 1.20.
We consider the growth of this metric locally near points a ∈ D . Here, we carry out only the case where a lies above the cusp ∞. The other cusps are treated analogously, and the easier case where a lies above an elliptic fixed point is left to the reader.
So, let a ∈ π * (∞), and let U ⊂ X( ) be a small open neighborhood of a such that π(U ) ⊂ V C with C > m/D, as in the discussion preceding Lemma 2.1. In view of the proof of Lemma 2.14, the function π * ( ∞ m ) n has precisely the divisor π * (nT (m)) on U . Thus s = 1 O(π * (nT (m))) /π * ( ∞ m ) n is a trivializing section for O π * (nT (m)) on U . By means of (2.46), we find that log s = n ϕ m (1) 2 π * log(y 1 y 2 ) + smooth function.
In the proof of Proposition 2.5, we already saw that π * (log(y 1 y 2 )) is a pre-log-log form on U . Hence the assertion follows from Lemma 1.21.

Notation 2.18
To lighten the notation, we frequently drop the π * . We write and also T (m) instead of π * T (m).

Remark 2.19
The Chern form ω m is computed and studied in [Br1,Theorem 7]. It turns out that Here, f (z 1 , z 2 ) is a certain Hilbert cusp form of weight 2 for K , essentially the Doi-Naganuma lift of the mth Poincaré series in S + 2 (D, χ D ). Green functions like G m are investigated in the context of the Weil representation in [BFu] and in the context of the theory of spherical functions on real Lie groups in [OT].

Star products on Hilbert modular surfaces
Here, we compute star products on Hilbert modular surfaces related to Hirzebruch-Zagier divisors. Throughout, let ≤ K be a subgroup of finite index.

Star products for Hirzebruch-Zagier divisors
In general, the product of a mixed growth form as G m and a pre-log-log form as ω 2 need not be integrable. Therefore the following lemma, which is crucial for Theorem 3.3, is special for Hilbert modular surfaces. It seems to be related to the Koecher principle.

Proof
By possibly replacing by a torsion-free subgroup of finite index, we may assume that \H 2 is regular. Since X( ) is compact, it suffices to show that G m is locally integrable in a neighborhood of any point of X( ). Outside the exceptional divisor D , this easily follows from the fact that G m has only logarithmic singularities along T (m) on \H 2 . Hence we only have to show local integrability at D . For simplicity, here, we treat the points only above the cusp ∞; for the other cusps, one can argue analogously.
We prove the assertion only in the case where κ is a cusp, leaving the other easier case to the reader. By possibly interchanging X( ) with an embedded desingularization of T (m 1 ) in X( ), we may assume that π * (∞), π * T (m 1 ), and π * (∞) ∪ π * T (m 1 ) are divisors with normal crossings. Since X( ) is compact, it suffices to show that (3.5) and (3.6) hold locally. We do this only for the cusp ∞; at the other cusps, one can argue analogously. Let a ∈ π * (∞) be a point on the exceptional divisor over ∞, and let U ⊂ X( ) be a small open neighborhood of a such that (π * T (m 2 )) ∩ U = ∅. After a biholomorphic change of coordinates, we may assume that U = E 2 , a = (0, 0), and π * T (m 1 ) = div(q α 1 q β 2 ) on U . We assume the notation of the proof of Lemma 3.1. Without loss of generality, it suffices to show that for some 1 ≥ δ > 0, We use the local expansions of G m 1 and G m 2 at the cusp ∞ given in (2.46). On E 2 , we have π * (G m 1 ) = − ϕ m 1 (1) 2 π * log(y 1 y 2 ) − log|q α 1 q β 2 | + smooth function, π * (G m 2 ) = − ϕ m 2 (1) 2 π * log(y 1 y 2 ) + smooth function.
We now estimate the integrals in (3.7) and (3.8). We consider only the case where both µ 1 and µ 2 are nonzero, leaving the easier case where one of them vanishes to the reader. Only the dr 1 dρ 1 dρ 2 -component of the integrand gives a nonzero contribution. After a calculation, we find that in (3.7), this component is bounded by dr 1 dρ 1 dρ 2 r 1 | log r 1 + log r 2 | 2 + r 1 + r 2 | log r 1 + log r 2 | dr 1 dρ 1 dρ 2 .

THEOREM 3.3 Let T (m 1 ), T (m 2 ), and T (m 3 ) be Hirzebruch-Zagier divisors such that all possible intersections on X( ) among them are proper and such that T (m 2 ) is anisotropic. Then
g(m 2 ) * g(m 3 ). (3.9) Here, T (m 1 ) denotes the strict transform of the divisor T (m 1 ) on X( ).

ᮀ
Notice that in the above star product, the components of the desingularization of the divisor T (m 1 ) do not contribute. In other words, this star product is independent of the choice we made in the desingularization and depends only on the Baily-Borel compactification X( ).

Remark 3.4
The formula for the star product in Theorem 3.3 agrees with the formula by Gillet and Soulé for the star product of Green currents when formally applied to the Baily-Borel compactification X( ) that is a singular space.

Remark 3.5
Since T (m) and g(m) are invariant under the full Hilbert modular group K , we clearly have

Integrals of Green functions
The purpose of this section is to compute the first integral in (3.9) in the case of ω m 2 = ω m 3 = 2πiω. Recall our normalization (2.14) of the invariant Kähler form ω.
We consider the quadratic space V and the lattices L = L(O K ), L = L (O K ) defined at the beginning of Section 2.3. We put W = 0 −1 1 0 . The Hilbert modular group K acts on L by γ.A = γ Aγ t for γ ∈ K . For every A = a ν ν b ∈ L , the graph {(W Az, z); z ∈ H} defines a divisor in H 2 given by the equation az 1 z 2 + νz 1 + ν z 2 + b = 0. The stabilizer K,A of A under the action of K can be viewed as an arithmetic subgroup of SL 2 (R) with finite covolume (see [Z1,Section 1], [G,Chapter 5.1]). By reduction theory, the subset of elements with norm m/D decomposes into finitely many K -orbits. The divisor T (m) on K \H 2 is given by Here, {±1} acts on L m by scalar multiplication. We may rewrite m (z 1 , z 2 , s) using this splitting. For A = a ν ν b ∈ L m , we define d A (z 1 , z 2 ) = 1 + |z 1 − W Az 2 | 2 2 (z 1 ) (W Az 2 ) = 1 + |az 1 z 2 + νz 1 + ν z 2 + b| 2 2y 1 y 2 m/D . (3.13) If γ ∈ K , we have This follows from the fact that |z 1 − z 2 | 2 /(2y 1 y 2 ) is a point-pair invariant; that is, it depends only on the hyperbolic distance of z 1 and z 2 . Consequently, (3.14) PROPOSITION 3.6 The integral converges for all s ∈ C with (s) > 1.

Proof
According to (3.14), we have the formal identity (3.15) By a standard Fubini-type lemma on integrals over Poincaré series (see, e.g., [Fr3, Appendix 2, Theorem 7]), the integral on the left-hand side converges (and equals the right-hand side) if the latter integral on the right-hand side converges. Thus it suffices to prove that converges. We notice that the inner integral actually does not depend on z 2 and A.
Using the fact that |z 1 − z 2 | 2 /(2y 1 y 2 ) is a point-pair invariant and the invariance of η 1 , we find that (3.16) is equal to This integral is obviously bounded by That the latter integral is finite for (s) > 1 is a well-known fact (see, e.g., [L, Chapter 14, Section 3]).

LEMMA 3.7
Let h : H → C be a bounded eigenfunction of the hyperbolic Laplacian 1 with eigenvalue λ. Then for s ∈ C with (s) > 1, we have

Proof
This statement is well known. It can be proved using the Green formula (see, e.g., [I,Chapter 1.9]; notice the different normalization there).
ᮀ THEOREM 3.8 Let f : K \H 2 → C be a bounded eigenfunction of the Laplacian 1 (or 2 ) with eigenvalue λ. Then for s ∈ C with (s) > 1, we have
First, we notice that the integral I on the left-hand side converges by Proposition 3.6. Similarly, as in the proof of Proposition 3.6, we rewrite it as Here, the inner integral can be computed by means of Lemma 3.7. We obtain This concludes the proof of the theorem. (3.18)

Proof
If we use Theorem 3.8 with f = 1, we get By (2.19), we have, in addition, Since m (z 1 , z 2 ) = lim s→1 ( m (z 1 , z 2 , s) − ϕ m (1)/(s − 1)) is regular and integrable at s = 1, we derive the second claim by comparing residues in the latter two equalities. The third claim follows from Definition 2.13 and (2.42) by comparing the constant terms.

Star products on isotropic Hirzebruch-Zagier divisors
For the rest of this section, we assume that p is a prime that is split in O K or p = 1. Let p be a prime ideal of O K above p. There are a fractional ideal c and a totally positive λ ∈ K such that p = λc 2 . We fix a matrix M ∈ c −1 c −1 c c ∩ SL 2 (K). It is well known that the isotropic Hirzebruch-Zagier divisor T (p) ⊂ K \ H 2 is irreducible (see [G,Chapter 5.1] and [HZ]). It may have points of self-intersection, and its normalization is isomorphic to the noncompact modular curve Y 0 (p) = 0 (p) \ H. The normalization of the closure of T (p) in X( K ) is isomorphic to the compact modular curve X 0 (p), the standard compactification of Y 0 (p) (for basic facts on integral models of X 0 (p), the line bundle of modular forms on it, and the normalization of the corresponding Petersson metric, we refer to [Kü2]).
We now describe how T (p) can be parametrized; later, in Proposition 5.11, we give a modular description of this map. On the Hilbert modular surface X( (O K ⊕p)), the Hirzebruch-Zagier divisor T p (p) is simply given by the diagonal. More precisely, the assignment τ → (τ, τ ) induces a morphism of degree 1, whose image is T p (p). In fact, the vector 0 . Moreover, it is easily checked that the stabilizer in (O K ⊕ p) of the diagonal in H 2 is equal to 0 (p). The cusp ∞ (resp., 0) of X 0 (p) is mapped to the cusp ∞ (resp., 0) of X( (O K ⊕ p)).

Proof
The map ϕ is given by the commutative diagram where the horizontal arrow is given by (3.20) and the vertical arrow (which is an isomorphism) by (2.5). The properties of the latter two maps imply the assertion. ᮀ It is easily seen that the pullback of the Kähler form ω equals 2 dx dy 4πy 2 . For the next proposition, we recall that the Fricke involution W p on the space of modular forms of weight k for 0 (p) is defined by f (z) → (f | k W p )(z) = p k/2 z −k f (−1/pz). Moreover, we define the Petersson slash operator for Hilbert modular forms in weight k by PROPOSITION 3.11 (i) If F is a Hilbert modular form of weight k for K , then its pullback is a modular form of weight 2k for the group 0 (p).

(ii)
If F is, in addition, holomorphic and has the Fourier expansions at the cusps c and c , respectively, then the Fourier expansions of ϕ * F at ∞ and 0 are given by (3.23) Remark 3.12 Proposition 3.11 implies, in particular, that Hilbert modular forms with rational Fourier coefficients are mapped to modular forms for 0 (p) with rational Fourier coefficients. This shows that ϕ is actually defined over Q (see Proposition 5.5).
We now compute certain star products of pullbacks of Hilbert modular forms via ϕ.
THEOREM 3.13 Let F , G be Hilbert modular forms of weight k with rational Fourier coefficients. Assume that all possible intersections on X( K ) of T (p), div (F ), div (G) are proper, and assume that F does not vanish at the cusps c and c −1 of K . Then Here, (div(ϕ * F ), div(ϕ * G)) X 0 (p),fin denotes the intersection number at the finite places on the minimal regular model X 0 (p) of X 0 (p) of the divisors associated with the sections of the line bundle of modular forms corresponding to ϕ * F and ϕ * G (see [Kü2]).

Star products on Hilbert modular surfaces
We combine the results of the previous sections to compute star products on Hilbert modular surfaces. THEOREM 3.14 Let p be a prime that is split in O K or p = 1, and let F , G be Hilbert modular forms of weight k with rational Fourier coefficients. Assume that all possible intersections on X( K ) of T (p), div (F ), div (G) are proper, and assume that F does not vanish at any cusp of K . Then

Proof
The logarithm of the Petersson norm of a Hilbert modular form satisfies along D the same bounds as the Green functions G m . Hence we may calculate the star product in question by means of the formula of Theorem 3.3: By Corollary 3.9(iii) and (2.33), the first integral is given by Here, we have used the fact that χ D (p) = 1. For the remaining integral, we use the morphism ϕ defined by (3.20) to infer where the last equality was derived by means of Theorem 3.13. Adding the above expressions, the claim follows by the identity ζ K (s) = ζ (s)L(s, χ D ).

Borcherds products on Hilbert modular surfaces
It was shown in [Br1,Section 4] that for certain integral linear combinations of the G m (z 1 , z 2 ), all Fourier coefficients, whose index ν ∈ d −1 has negative norm, vanish.
Such a linear combination is then the logarithm of the Petersson metric of a Hilbert modular form, which has a Borcherds product expansion. We now explain this in more detail.

Basic properties of Borcherds products
Recall our assumption that D be a prime. Let k be an integer. We denote by A k (D, χ D ) the space of weakly holomorphic modular forms of weight k with character χ D for the group 0 (D). These are holomorphic functions f : H → C, which satisfy the transformation law , and are meromorphic at the cusps of 0 (D). If f = n∈Z c(n)q n ∈ A k (D, χ D ), then the Fourier polynomial n<0 c(n)q n is called the principal part of f . Here, q = e 2πiτ , as usual. We write M k (D, χ D ) (resp., S k (D, χ D )) for the subspace of holomorphic modular forms (resp., cusp forms). For ∈ {±1}, we let A k (D, χ D ) be the subspace of all f = n∈Z c(n)q n in A k (D, χ D ) for which c(n) = 0 if χ D (n) = − (see [BB]). A classical lemma due to Hecke implies that (see, e.g., [O,Lemma 6,p. 32]). Finally, we define the spaces M k (D, χ D ) and S k (D, χ D ) analogously.
Here, we mainly consider M + 2 (D, χ D ) and A + 0 (D, χ D ). The Eisenstein series is a special element of M + 2 (D, χ D ). Note that by (3.17), The space M + 2 (D, χ D ) is the orthogonal sum of CE and the subspace of cusp forms S + 2 (D, χ D ). The existence of weakly holomorphic modular forms in A + 0 (D, χ D ) with the prescribed principal part is dictated by S + 2 (D, χ D ). Before making this more precise, it is convenient to introduce the following notation. If n∈Z c(n)q n ∈ C((q)) is a formal Laurent series, we put We now recall [BB,Theorem 6], which is a reformulation of [B2, Theorem 3.1]. By Borcherds's theory [B1,Theorem 13.3], there is a lift from weakly holomorphic modular forms in A + 0 (D, χ D ) to Hilbert modular forms for the group K , whose divisors are linear combinations of Hirzebruch-Zagier divisors. Since this result is vital for us, we state it in detail. THEOREM 4.3 (see [B1,Theorem 13.3], [Br1,Theorem 5], [BB,Theorem 9]) Let f = n∈Z c(n)q n ∈ A + 0 (D, χ D ), and assume thatc(n) ∈ Z for all n < 0. Then there is a meromorphic function F (z 1 , z 2 ) on H 2 with the following properties.
F is a meromorphic modular form for K with some multiplier system of finite order. The weight of F is equal to the constant coefficient c(0) of f . It can also be computed using Theorem 4.1.
The divisor of F is determined by the principal part of f . It equals

S(−n),
and define the "Weyl vector" ρ W ∈ K for W and f by (W,−n) λ. (4.6) The function F has the Borcherds product expansion The product converges for all (z 1 , z 2 ) with y 1 y 2 > | min{n; c(n) = 0}|/D outside the set of poles. (iv) The Petersson metric of F is given by (4.7) (v) We have

Proof
The statements (i), (ii), and (iii) are proved in [BB] using [B1,Theorem 13.3]. Therefore we only have to verify (iv) and (v). By Theorem 4.1, the existence of f ∈ A + 0 (D, χ D ) implies that condition (4.5) is fulfilled for all cusp forms g ∈ S + 2 (D, χ D ). Using Poincaré series, it is easily checked that (4.5) actually holds for all g ∈ S 2 (D, χ D ). Thus, by [Br1,Theorem 5], the right-hand side of (4.7) is equal to the logarithm of the Petersson metric of a Hilbert modular form F with the same divisor as F . Hence the quotient F /F is a Hilbert modular form without any zeros and poles on H 2 and thereby constant. This shows that (4.7) holds up to an additive constant. By comparing the constant terms in the Fourier expansions of both sides, one finds that this constant equals zero. Here, the Fourier expansion of the right-hand side is given by (2.46). This proves (iv). The last assertion follows from (iv) and (3.18) in Corollary 3.9. ᮀ Notice that in [Br1], the assertions (i) and (ii) are deduced from (iv). There, however, a slightly different product expansion is obtained, which involves Fourier coefficients of weakly holomorphic Poincaré series of weight 2. Similarly, as in [Br2,Chapter 1], these can be related to the coefficients of weakly holomorphic modular forms of weight zero. In that way, a more direct proof of Theorem 4.3 can be given. Another direct proof can be obtained by completely arguing as in [Br2]. There, the Green functions m (z 1 , z 2 , s) are constructed as regularized theta lifts of nonholomorphic Hejhal-Poincaré series of weight zero. Here, for brevity, we have preferred to argue as above. Observe that (v) also follows from [Ku5,Main Theorem 2.12].

Definition 4.4
Hilbert modular forms that arise as lifts via Theorem 4.3 are called Borcherds products. A holomorphic Borcherds product is called integral if it has trivial multiplier system and integral coprime Fourier coefficients. A meromorphic Borcherds product is called integral if it is the quotient of two holomorphic integral Borcherds products.

PROPOSITION 4.5 For any Borcherds product F , there exists a positive integer N such that F N is integral. Proof
Let f = n∈Z c(n)q n ∈ A + 0 (D, χ D ), as in Theorem 4.3, be the preimage of F under the Borcherds lift. It is explained in [BB,Proposition 8] that the conditionc(n) ∈ Z for n < 0 automatically implies that all coefficients c(n) of f are rational with bounded denominators.
Thus, if F is holomorphic, the Borcherds product expansion of F implies that a suitable power of F has integral coprime Fourier coefficients. Since the multiplier system of F has finite order, we obtain the assertion in that case.
It remains to show that any meromorphic Borcherds product is the quotient of two holomorphic Borcherds products. In view of Theorem 4.3(ii), it suffices to show that there exist two weakly holomorphic modular forms f j = n∈Z c j (n)q n ∈ A + 0 (D, χ D ) such thatc j (n) ∈ Z ≥0 for all n < 0 (where j = 1, 2) and f = f 1 − f 2 . Then F is the quotient of the Borcherds lifts of f 1 and f 2 . We now construct such forms explicitly.

Density of Borcherds products Definition 4.7
Intersection points z ∈ K \H 2 of Hirzebruch-Zagier divisors are called special points (see [G,Chapter 5.6

]).
Particular examples of special points are the elliptic fixed points (if D > 5). THEOREM 4.8 If S ⊂ K \H 2 is a finite set of special points, then there exist infinitely many meromorphic Borcherds products of nonzero weight whose divisors are disjoint from S and are given by linear combinations of isotropic Hirzebruch-Zagier divisors T (p) with p prime and coprime to D.
Here and in the following, by "infinitely many Borcherds products" we understand infinitely many Borcherds products whose divisors have pairwise proper intersection.
To prove this theorem, it is convenient to view SL 2 (O K ) as an orthogonal group. We briefly recall some facts on the identification of (SL 2 (R)×SL 2 (R))/{±(1, 1)} with the group SO 0 (2, 2)/{±1} (for more details, see [G,Chapter 5.4 (L). In terms of Gr(L), the Hirzebruch-Zagier divisor T (m) is given by where λ ⊥ means the orthogonal complement of λ in Gr (L).
For v ∈ Gr(L), we denote by L v the lattice L ∩ v ⊥ with the integral quadratic form q v = D · q| L v . If g ∈ O(L), then the quadratic modules (L v , q v ) and (L gv , q gv ) are equivalent. Therefore the equivalence classes of these quadratic forms can be viewed as invariants of the points of K \H 2 . The following lemma is well known. LEMMA 4.9 Let z ∈ K \H 2 , and assume that z corresponds to v ∈ Gr (L). Then z ∈ T (m) if and only if the quadratic form q v on L v represents m.
It is easily checked that L v has rank 2 if and only if v corresponds to a special point z ∈ H 2 . In this case, (L v , q v ) is a positive definite integral binary quadratic form. If we write v for its discriminant, then v < 0 and v ≡ 0, 1 (mod 4). LEMMA 4.10 If S ⊂ K \H 2 is a finite set of special points, then there are infinitely many primes p coprime to D such that T (p) is nonempty, isotropic, and T (p) ∩ S = ∅.

Proof
Let Q 1 , . . . , Q r be the positive definite integral binary quadratic forms corresponding to the special points in S. In view of the above discussion, it suffices to show that there exist infinitely many primes p, which are not represented by Q 1 , . . . , Q r , and such that χ D (p) = 1.
Let j be the discriminant of Q j . Then j < 0 and j ≡ 0, 1 (mod 4). Let n be a nonzero integer coprime to j . It is well known that R( j , n), the total representation number of n by positive definite integral binary quadratic forms of discriminant j , is given by (see [Z2,Section 8]). Hence, if R( j , n) = 0 for n coprime to j , then Q j does not represent n. In particular, any prime p with χ j (p) = −1 is not represented by Q j .
Thus it suffices to show that there are infinitely many primes p with χ j (p) = −1 (j = 1, . . . , r), Since the j are negative and D is positive, this is clearly true. In fact, even a positive proportion of primes has these properties.

ᮀ
For the rest of this section, we temporarily abbreviate M := M + 2 (D, χ D ) and S := S + 2 (D, χ D ). We denote the dual C-vector spaces by M ∨ and S ∨ , respectively. For any positive integer r, the functional is a special element of M ∨ , and M ∨ is generated by the family (a r ) r∈N as a vector space over C. We denote by M ∨ Z the Z-submodule of M ∨ generated by the a r (r ∈ N). The fact that M has a basis of modular forms with integral coefficients implies that the rank of M ∨ Z equals the dimension of M and that M ∨ Z ⊗ Z C = M ∨ . We writē for the natural map given by the restriction of a functional. LEMMA 4.11 Let I be an infinite set of positive integers m with χ D (m) = −1, and let A ∨ be the Z-submodule of M ∨ Z generated by the a m with m ∈ I . Then there is a nonzero a ∈ A ∨ with the property thatā = 0 in S ∨ .
If the corresponding linear combination of the coefficients of the Eisenstein series E does not vanish, then a = r 0 (m)a m + r 1 (m)a n 1 + · · · + r d (m)a n d is a nonzero element of A ∨ with the claimed property, and we are done. We now assume that the linear combination (4.9) vanishes for all m ∈ I and derive a contradiction. If the vector r(m) = (r 1 (m), . . . , r d (m)) is equal to zero for some m, then the vanishing of (4.9) implies that r 0 (m) = 0, contradicting our assumption on the r j (m). Therefore we may further assume that r(m) = 0 for all m ∈ I . Equation (4.8) and the vanishing of (4.9) imply B D (m) r 1 (m)ā n 1 + · · · + r d (m)ā n d = r 1 (m)B D (n 1 ) + · · · + r d (m)B D (n d ) ā m (4.10) for all m ∈ I . We write r for the Euclidean norm of a vector r = (r 1 , . . . , r d ) ∈ C d and also denote by · a norm on S ∨ , say, the operator norm. Sinceā n 1 , . . . ,ā n d are linearly independent, there exists an ε > 0 such that r 1ān 1 + · · · + r dān d ≥ ε r for all r ∈ C d . Moreover, there exists a C > 0 such that |r 1 B D (n 1 ) + · · · + r d B D (n d )| ≤ C r for all r ∈ C d . If we insert these estimates into the norm of (4.10), we obtain Since r(m) = 0, we find that for all m ∈ I . By We may remove the primes p with c(p) = 0 from the set I and repeat the above construction to get another Borcherds product. By induction, we get infinitely many Borcherds products with the claimed properties. ᮀ THEOREM 4.12 Let C ∈ Div(X( K )) be a linear combination of Hirzebruch-Zagier divisors. Then there exist infinitely many meromorphic integral Borcherds products F 2 and F 1 of nonzero weights such that (i) all possible intersections of div(F 1 ), div (F 2 ), and C are proper; (ii) F 2 (κ) = 1 at all cusps κ of X( K ); (iii) div (F 1 ) is a linear combination of isotropic Hirzebruch-Zagier divisors of prime discriminant p coprime to D (and thus χ D (p) = 1).

Proof
Since there are infinitely many anisotropic Hirzebruch-Zagier divisors, Theorem 4.1 and Lemma 4.11 imply that there are infinitely many Borcherds products F 2 of nonzero weight, whose divisor consists of anisotropic Hirzebruch-Zagier divisors, and such that C and div(F 2 ) intersect properly. Choose such an F 2 . Since F 2 has an anisotropic divisor, the Weyl vectors in the Borcherds product expansion of F 2 at the different cusps of X( K ) equal zero. Consequently, F 2 is holomorphic at all cusps with value 1. Let S ⊂ K \H 2 be the finite set of intersection points div(F 2 ) ∩ C. By Theorem 4.8, there exist infinitely many Borcherds products F 1 of nonzero weight such that div (F 1 ) is disjoint to S and such that properties (i) and (iii) are fulfilled.
By possibly replacing F 2 , F 1 by sufficiently large powers, we may assume that these are integral Borcherds products.

ᮀ
In the rest of this section, we essentially show that the subspace of Pic(X( K )) ⊗ Z Q, spanned by all Hirzebruch-Zagier divisors, is already generated by the T (p) with prime index p and χ D (p) = 1. Proof Since M has a basis of modular forms with integral coefficients, it suffices to show that the a p with p ∈ I generate M ∨ as a C-vector space. In view of Lemma 4.11, it suffices to show that theā p generate S ∨ . Therefore the assertion is a consequence of the following lemma. ᮀ LEMMA 4.14 Let I be as in Proposition 4.13. If f ∈ S is a cusp form that is annihilated by allā p with p ∈ I , then f = 0.

Proof
Since S has a basis of modular forms with rational Fourier coefficients, we may assume, without loss of generality, that the Fourier coefficients of f are algebraic. By the hypothesis, and because f ∈ S = S + 2 (D, χ D ), we have a p (f ) = 0 for almost all primes p.
The assertion follows from the properties of the -adic Galois representations associated to a basis of normalized newforms of S using the main lemma of [OS] (by a similar argument as on [OS,p. 461]). Notice that S does not contain any eigenforms with complex multiplication since D is a prime ≡ 1 (mod 4). Here,c(p) = 0 for all but finitely many p ∈ I . Therefore, in view of Theorem 4.1, there exists a weakly holomorphic modular form f ∈ A + 0 (D, χ D ) with the principal part c(m)q −m − p∈I c(p)q −p and vanishing constant term c(0). The Borcherds lift of f is a Borcherds product of weight zero with divisor of the required type. By taking a sufficiently large power, we may assume that it is integral. We may now remove the primes p occurring with c(p) = 0 in the above sum from the set I and repeat the argument. Inductively, we find infinitely many Borcherds products of the required type. Proof By Theorem 4.8, we can find infinitely many integral meromorphic Borcherds products G of nonzero weight such that div(G) is a linear combination of isotropic Hirzebruch-Zagier divisors T (p) with p prime and coprime to mD. In particular, div (G) does not have T (m) as a component. The product of any such G and any F as in Theorem 4.15 is a Borcherds product with the required properties.

Arithmetic theory of Hilbert modular surfaces
Throughout this section, we keep the assumptions of the previous sections. In particular, D is a prime congruent 1 modulo 4 and K = Q( √ D).

Moduli spaces of abelian schemes with real multiplication
In this section, we recall some background material on integral models of Hilbert modular surfaces. The reader is referred to [R], [DP], [P], and [V] for more details.
Suppose that A → S is an abelian scheme. Then there exist a dual abelian scheme A ∨ → S and a natural isomorphism A ∼ = (A ∨ ) ∨ . If φ : A → B is a homomorphism of abelian schemes, then there is a dual morphism φ ∨ : In this case, we write µ = µ ∨ . We denote by Hom(A, A ∨ ) sym the space of symmetric homomorphisms.
An abelian scheme A → S of relative dimension 2, together with a ring homomorphism is called an abelian surface with multiplication by O K and denoted by the pair (A, ι). Via α → ι(α) ∨ , we obtain an O K -multiplication on the dual abelian surface. If a ⊂ O K is an ideal, we write A[a] for the a-torsion on A.
Suppose from now on that (A, ι) is an abelian surface with multiplication by We denote by P (A) the sheaf for theétale topology (largeétale site) on Sch /S defined by for all T → S. We write P (A) + for the subsheaf of polarizations in P (A). The pair (A, ι) is said to satisfy the Deligne-Pappas condition (DP) if the canonical morphism of sheaves is an isomorphism. In this case, P (A) is a locally constant sheaf of projective O Kmodules of rank 1 (see [V,Proposition 1.4]). The Deligne-Pappas condition (DP) is, over Z[1/D]-schemes, equivalent to the Rapoport condition (R) that 1 A/S be locally on S a free (O S ⊗ Z O K )-module. Moreover, in characteristic zero, it holds automatically (see, e.g., [Go,p. 99]).
Let l be a fractional ideal of K, and let l + be the subset of totally positive elements. An l-polarization on (A, ι) is a homomorphism of O K -modules ψ : l −→ P (A) S , taking l + to P (A) + S , so that the natural homomorphism is an isomorphism. By [V,Proposition 3.3 between the constant group scheme defined by (O K

/NO K ) 2 and the N-torsion on A.
With the formulation of the next theorem, which summarizes some properties of the moduli spaces of abelian surfaces needed below, we follow [P,Theorem 2.1.2,p. 47;Remark 2.1.3,p. 47] (see also [C], [R], [Go]; in particular, see [Go,Theorem 2.17,p. 57;Lemma 5.5,p. 99]). Furthermore, we choose a primitive Nth root of unity ζ N .
The moduli problem "Abelian surfaces over S with real multiplication by O K and l-polarization with (DP) and full level-N structure" is represented by a regular algebraic stack H l (N), which is flat and of relative dimension 2 over Spec Z[ζ N , 1/N]. It is smooth over Spec Z[ζ N , 1/ND], and the fiber of H l (N) above D is smooth outside a closed subset of codimension 2. Moreover, if N ≥ 3, then H(N) is a scheme.
It is well known that (O K ⊕ a)\H 2 can be identified with H ad −1 (1) (C). The isomorphism can be described as follows (see, e.g., [Go,Chapter 2.2] for a detailed discussion).
To z = (z 1 , z 2 ) ∈ H 2 , we associate the lattice If γ = a b c d ∈ GL 2 (K) with totally positive determinant, then In particular, if γ ∈ (O K ⊕ a), then γ z = (1/(cz + d)) z . For any r ∈ ad −1 , we define a hermitian form on C 2 by For αz + β, γ z + δ ∈ z ⊂ C 2 , we have The hermitian form H r,z is positive definite (and therefore defines a polarization) if and only if r ∈ (ad −1 ) + . We see that We write λ r for the O K -linear homomorphism in P (A z ), given by x → H r,z (x, ·). The assignment r → λ r defines an O K -linear isomorphism ψ : ad −1 → P (A z ), which maps the totally positive elements to polarizations.
Observe that the moduli algebraic stack H l (N) depends (up to a canonical isomorphism) only on the ideal class of l. To lighten the notation, we frequently omit the superscript l whenever l ∼ = d −1 ; for example, we simply write H(N) for H l (N). We write H = H(1). By abuse of notation, we denote the (coarse) moduli schemes associated with H and H(N) by H and H(N), respectively.

THEOREM 5.2
There is a toroidal compactification h N : which is smooth at infinity and such that forgetting the level induces a morphism π N : H(N) → H(1) which is a Galois cover. The complement H(N) \ H(N) is a relative divisor with normal crossings.
We also refer to [C] and [R] for the fact that one can choose a compatible system of toroidal compactifications for which we have morphisms π M,N : H(M) → H(N) whenever N|M.

THEOREM 5.3 (q-expansion principle)
There is a positive integer n 0 (depending on K and N) such that in all weights k divisible by n 0 , there exists a line bundle M k ( K (N)) on H(N), whose global sections correspond to holomorphic Hilbert modular forms of weight k for K (N) with Fourier coefficients in Z[ζ N ,1/N]. If N ≥ 3, then we can put n 0 = 1.
We call M k ( K (N)) the line bundle of Hilbert modular forms. It is given by the kth power of the Hodge bundle, that is, the pullback along the zero section of the determinant of the relative cotangent bundle of the universal family over H(N).
According to Proposition 4.5, any integral Borcherds product of weight k (divisible by n 0 ) defines a rational section of M k ( K (N)).

PROPOSITION 5.5
For any m, the divisor T (m) ⊂ X( K ) = H(C) is defined over Q.

Proof
This fact is well known (see, e.g., [HLR], [G] for related formulations). We sketch a different proof using the theory of Borcherds products. Let p be a prime that is split in O K . According to Proposition 3.10, T (p) is the image of the morphism ϕ defined in (3.20). By Proposition 3.11, the pullback via ϕ of a Hilbert modular form with rational Fourier coefficients is a modular form for 0 (p) with rational coefficients. Thus the ideal of holomorphic Hilbert modular forms vanishing along T (p) in the graded ring of holomorphic Hilbert modular forms is generated by Hilbert modular forms with rational coefficients. Since X( K ) is equal to Proj n 0 |k H 0 ( H, M k ( K ))(C), we obtain the claim for T (p) (see also Proposition 5.11).
If T (m) is any Hirzebruch-Zagier divisor, then by Theorem 4.15, there exists an integral Borcherds product whose divisor is a linear combination of T (m) and divisors T (p) with prime index p, as above. Now the claim follows by linearity. If N = 1, we frequently write T(m) for T 1 (m). Observe that T N (m) = π * N T(m).

Proof
Without loss of generality, we may assume that F is holomorphic. It suffices to show that div N (F ) is a horizontal divisor. Since F is a modular form for the full group K , the divisor div N (F ) is the pullback of a divisor on H. Because H is geometrically irreducible at all primes (see [DP,p. 65]), the vertical part of div N (F ) can contain only multiples of full fibers of H(N). But the Borcherds product expansion implies that the Fourier coefficients of F are coprime. Therefore, by the q-expansion principle (see [C,Theorem 4.1]), div N (F ) does not contain a full fiber of H(N) above Spec Z[ζ N , 1/N].
This concludes the proof of the proposition.

Modular morphisms
We extend the morphism ϕ : Y 0 (p) → K \H 2 of Section 3.3 to integral models by giving a modular interpretation. We basically follow the descriptions in [Go,Chapter 2.5.1] and [La,p. 134] (see also [KR, Remark (ii), p. 169] for related results). Moreover, we extend it to compactifications by means of the q-expansion principle.
For the rest of this section, we assume that p is a prime that is split in O K or p = 1. Let p be a prime ideal of O K above p. There are a fractional ideal c and a totally positive λ ∈ K such that p = λc 2 . We may assume that N(λ) is a power of p (e.g., if we take c = p (h+1)/2 , where h is the class number of K). We fix a matrix M ∈ c −1 c −1 c c ∩ SL 2 (K). The following two lemmas were communicated to us by T. Wedhorn [W]. Proof (i) We write ( ) p for ( ) ⊗ Z Z p . It suffices to show that the length of the cokernel of π * p : P (B) p → P (A) p is equal to the p-adic valuation of deg(π) for any prime number p that divides the degree of π. We have a commutative diagram where T p (A) and T p (B) denote the Tate modules corresponding to A and B, and for an alternating form of (O K [V, (1.7.4) we obtain the first claim.
(ii) Let C be any abelian scheme over S with O K -multiplication. We first claim that a Hom O K (A, C) ⊂ Hom O K (B, C)π. Let x ∈ a, and let α ∈ Hom O K (A, C). We have to show that xα annihilates A [a], but this is obvious as α is O K -linear.
Next, we claim that a . Note that if we dualize the canonical projection π : A → B, π ∨ is the canonical projection . This follows from the fact that the dual of the multiplication with an element x ∈ O K on any abelian scheme A with O K -multiplication is just the multiplication with x on A ∨ . Hence the second claim follows from the first one by dualizing.
Altogether, these two claims imply that a 2 Hom It follows that a 2 P (A) ⊂ π * P (B) ⊂ P (A), and as P (A) is a locally constant sheaf of projective O K -modules of rank 1, this is also true for P (B). Moreover, the degree of π is N(a) 2 . Hence the assertion follows from (i).   A[p] implies that both π and ξ have degree greater than 1. Since the degree of pr is equal to p 2 , we have deg(π) = deg(ξ ) = p. In view of Lemma 5.8, the above diagram induces the following inclusions for the polarization modules: Therefore π * P (B) = pP (A). ᮀ Let l be a fractional ideal of K, and let E/S be an elliptic curve over a scheme S. We consider the abelian surface with the canonical O K -action, the isomorphism being obtained by choosing a Z-basis to l. Then the natural O K -action on l is given by a ring homomorphism O K → M 2 (Z), u → R u . The corresponding O K -action on E × S E is given by the inclusion M 2 (Z) ⊂ End(E × S E). One easily checks that for any u ∈ O K , the conjugate acts by The dual of the abelian surface with O K -multiplication E ⊗ Z l is given by E ⊗ Z (ld) −1 . Moreover, P (E ⊗ Z l) ∼ = l −2 d −1 , the isomorphism preserves the positivity, and the Deligne-Pappas condition (DP) holds.

Proof
If we choose a Z-basis of l, the natural O K -action on l is given by a ring homomorphism The dual of l with respect to the trace form on K is equal to (ld) −1 , and the dual O K -action with respect to the dual basis is given by x → R t x . On the other hand, we may identify (E × S E) ∨ canonically with E × S E. This identifies the dual R ∨ of a morphism R ∈ M 2 (Z) ⊂ End(E × S E) with R t . This yields the first assertion.
We obtain an O K -linear monomorphism l −2 d −1 → P (E ⊗ Z l) by the assignment x → (e ⊗ l → e ⊗ xl). One can check that totally positive elements are mapped to polarizations. The natural composite morphism is an isomorphism. Now, the assertion follows from [V,Proposition 3.3 (DP). Moreover, the associated morphism Y 0 (p)(C) → H(C) is induced by the morphism of Section 3.3, H → H 2 , τ → M −1 λ 0 0 1 (τ, τ ).

Proof
Let S be a scheme over Z[1/p], let E be an elliptic curve over S, and let C be a finite locally free subgroup scheme of E whose geometric fibers are cyclic groups of order p. We consider A = E ⊗ Z c as in (5.4) with c as in the beginning of Section 5.2.
Since H satisfies the conditions of [AG,Corollary 3.2], the abelian scheme B satisfies (DP) for P (B), which is equal to pc −2 d −1 ∼ = d −1 by Lemmas 5.9 and 5.10. The assignment (E, C) → B is functorial and defines a morphism Y 0 (p) → H.
We now trace this morphism on the complex points. Recall that points on 0 (p)\H correspond to isomorphism classes of elliptic curves over C together with a subgroup of order p via the assignment τ → = (E τ , C τ ) := (C/ τ , 1/p ), where τ = Zτ +Z, and 1/p denotes the subgroup of E τ generated by the point 1/p + τ . The abelian surface A is defined by the lattice τ,τ = c(τ, τ ) + c. The lattice for the quotient A/(C τ ⊗ c) is given by c(τ, τ ) + (1/p)c ⊂ C 2 and that for A/A[p] is given by cp −1 (τ, τ ) + cp −1 . Therefore the lattice for B is equal to B = c(τ, τ ) + cp −1 . On the other hand, the abelian surface corresponding to the point M −1 λ 0 0 1 z ∈ K \H 2 is given by the lattice˜ z = O K  (5.2), we see that˜ Therefore B ∼ =˜ (τ,τ ) , and the associated morphism we obtain a moduli description of the morphism of (3.19). (ii) If we simply consider E → A = E ⊗ Z O K (and do not take a quotient), we obtain a moduli description of the diagonal embedding SL 2 (Z)\H → K (see [Go,Chapter 2.5]).
By abuse of notation, we denote the coarse moduli scheme associated with Y 0 (p) by Y 0 (p), too. In the following proposition, we extend the morphism of moduli schemes Y 0 (p) → H given by Proposition 5.11 to the minimal compactifications. Recall that the minimal compactification X 0 (p) of Y 0 (p) can be described analogously to (5.3) as X 0 (p) = Proj k≥0, n 1 |k H 0 X 0 (p), M k ( 0 (p)) . (5.5) Here, n 1 is a suitable positive integer (depending on p), and (for k divisible by n 1 ) M k ( 0 (p)) denotes the line bundle of modular forms on X 0 (p). Moreover, by the q-expansion principle on modular curves, the global sections of the bundle M k ( 0 (p)) correspond to modular forms of weight k for 0 (p) whose Fourier coefficients at the cusp ∞ belong to Z[1/p].

PROPOSITION 5.13
There exists a unique proper morphismφ : X 0 (p) → H of schemes over Z[1/p] such that the diagram Since taking q-expansions commutes with base change, the map on q-expansions given by (3.21) and (3.22) defines the desired homomorphismφ * over Z[1/p].
By [H,Exercise 2.2.14],φ * induces a morphism defined on some open subset U ⊂ Proj n 0 |k H 0 X 0 (p), M 2k ( 0 (p)) . By construction, U contains Y 0 (p). We now prove that U = X 0 (p). It suffices to show that there is an F ∈ S + such thatφ * (F ) does not vanish on the cusps 0 and ∞ as sections over Z[1/p]. By the q-expansion principle, it suffices to show that there is a holomorphic Hilbert modular F of positive weight with integral Fourier coefficients such that the constant coefficient ofφ * (F ) at ∞ (resp., 0) is a unit in Z[1/p]. Since the constant coefficient ofφ * (F ) at ∞ is up to a power of p equal to the constant coefficient of F at c (resp., the constant coefficient ofφ * (F ) at 0 is up to a power of p equal to the constant coefficient of F at c −1 ), it suffices to show that there is a holomorphic Hilbert modular form F of positive weight with integral Fourier coefficients whose constant coefficient at c (resp., c −1 ) is a unit in Z[1/p]. But the existence of such an F follows from [C,Proposition 4.5]. (Alternatively, we can use an integral Borcherds product whose divisor consists of anisotropic Hirzebruch-Zagier divisors.) The properness ofφ follows from [H,Corollary 2.4.8(e)]. By Proposition 5.11, we know that the Hirzebruch-Zagier divisor T (p) on the generic fiber H Q is equal to the image ofφ ⊗ Q. Sinceφ is proper, it follows that the Zariski closure of T (p) is contained in the image ofφ. The assertion follows from the irreducibility of X 0 (p). ᮀ Remark 5.14 The above argument, in particular, shows that for a Hilbert modular form F of weight k with rational coefficients, the pullback to X 0 (p) of the corresponding section of M k ( K ) over Z[1/p] is equal to the section of M 2k ( 0 (p)) over Z[1/p] corresponding to the pullbackφ * (F ) over C.

PROPOSITION 5.15
Let F , G be Hilbert modular forms for K with rational coefficients of weight k (with k divisible by n 0 and n 1 ). Assume that all possible intersections among div F , div G, and T (p) on X( K ) are proper. Let div N (F ), div N (G) be the divisors on H(N) of the rational sections of M k ( K (N)) associated with F and G. Moreover, let S = Spec Z[1/Np] and g : X 0 (p) → S be given by the structure morphism. Then we have, in CH 1 fin (S) = Z 1 (S),

Proof
In the following, we consider all schemes as schemes over S. We use the diagram (5.7) We may view F and G as sections of the line bundle of modular forms on H(N), H, and H. Throughout the proof, we temporarily denote the corresponding Cartier divisors by div N (F ), div 1 (F ), and div(F ), respectively (and analogously for G). So, div N (F ) = π * N div 1 (F ) and div 1 (F ) = u * div (F ). We may use intersection theory for the intersection of a Cartier divisor and a cycle as described in [Fu, [Fu,Chapter 20.1]).
By means of the projection formula, we obtain, in CH 2 For the latter equality, we have used the fact that π N is a flat morphism.
Combining this with (5.7) and Remark 5.14, we obtain the assertion.

Arithmetic intersection theory on Hilbert modular varieties
Since there exists no arithmetic intersection theory for the stack H(1), we work with the tower of schemes { H(N)} N≥3 as a substitute for H(1). We define in Section 6.3 the arithmetic Chow ring of the Hilbert modular variety without level structure to be the inverse limit of the arithmetic Chow rings associated to this tower. In this way, we obtain R-valued arithmetic intersection numbers; even so, for every level N, we need only calculate the arithmetic intersection numbers up to contributions from the finite places dividing N. We keep our assumption that K be a real quadratic field with prime discriminant D. Moreover, let N be an integer at least 3. Throughout this section, we mainly work over the arithmetic ring Z[ζ N , 1/N]. We put S N = Spec Z[ζ N , 1/N] and Consequently, we have an arithmetic degree map We Analogously, we let D ∞ = σ ∈ ( H(N) \ H(N)) σ (C). By construction, D ∞ is a normal crossing divisor on H(N) ∞ which is stable under F ∞ . We write D pre for the Deligne algebra with pre-log-log forms along D ∞ . These data give rise to arithmetic Chow groups with pre-log-log forms denoted by CH * ( H(N), D pre ).
We write for the total degree of the morphism H(N) → H of schemes over Z[1/N].
From Proposition 2.5, we deduce that the line bundle of modular forms of weight k on H(N) equipped with the Petersson metric is a pre-log singular hermitian line bundle, which we denote by M k ( K (N)). Its first arithmetic Chern class defines a class c 1 M k ( K (N)) ∈ CH 1 H(N), D pre which one may represent by (div (F ), g N (F )), where F is a Hilbert modular form of weight k with Fourier coefficients in Q(ζ N ) and g N (F ) = (2πik · ω, − log F σ ) σ ∈ . The first arithmetic Chern class c 1 M k ( K (N)) is linear in the weight k.

Arithmetic Hirzebruch-Zagier divisors
If we consider G m (z 1 , z 2 ) as a Green function on K (N)\H 2 , it immediately follows from Proposition 2.17 that g N (m) = −2∂∂G m (z 1 , z 2 ), G m (z 1 , z 2 ) σ ∈ is a Green object for the divisor T N (m) ∞ on H(N) ∞ . We obtain the following arithmetic Hirzebruch-Zagier divisors:

Proof
We choose for T N (−n) the representatives given in (6.2). That the subspace of CH 1 ( H(N), D pre ) Q spanned by the T N (m) has dimension at most dim C M + 2 (D, χ D ) follows by Theorem 4.1 arguing as in [B2,Lemma 4.4]. Here, one also needs Proposition 5.7 and its analytical counterpart Theorem 4.3(iv). On the other hand, the dimension is at least dim C M + 2 (D, χ D ) because of [Br1,Theorem 9], which says that any rational function on X( K ) whose divisor is supported on Hirzebruch-Zagier divisors is a Borcherds product.
Let I be a set of primes as above, and let T (m) be any Hirzebruch-Zagier divisor. By Theorem 4.15, there exist an integral Borcherds product of weight zero and thereby a rational function on H(N) with divisorc(m)T (m) + p∈Ic (p)T (p) on X( K ) and c(m) = 0. We may conclude by Proposition 5.7 and Theorem 4.3(iv) that This proves the theorem. in CH 1 ( H(N), D pre ) Q . Since A + 0 (D, χ D ) has a basis of modular forms with rational coefficients, it suffices to check (6.4) for those f withc(n) ∈ Z for n < 0. Then, by Theorem 4.3, there exists a Borcherds product F of weight c(0) with divisor n<0c (n)T (−n) on X( K ). We may assume that F is an integral Borcherds product and therefore defines a section of M c(0) ( K (N)). If we choose for T N (−n) the representatives of (6.2), we may conclude by Proposition 5.7 and Theorem 4.3(iv) that n<0c (n) T N (−n) = n<0c (n) T N (−n), g N (−n) = div N (F ), g N (F ) .
By (1.12), the right-hand side of the latter equality equals c 1 M c(0) ( K (N)) . Using the linearity of the arithmetic Chow groups, we obtain (6.4) and hence the assertion. ᮀ

Arithmetic intersection numbers and Faltings heights
Recall our convention that for an arithmetic cycle α ∈ CH 3 ( H(N), D pre ) Q , we frequently write α instead of deg((h N ) * α). LEMMA 6.3 Let k be a positive integer, and let p be a prime that is split in O K or p = 1; then there is a γ p,k ∈ Q such that we have, in R N , In the proof of Theorem 6.4, we show that γ p,k = 0 when (p, N) = 1.

Proof
Throughout the proof, all equalities that contain the image of the arithmetic degree map deg are equalities in R N . Without loss of generality, we may assume k sufficiently large. In order to prove the lemma, we represent T N (p) by (T N (p), g N (p)). We also choose integral Borcherds products F , G of nonzero weight k so that F (κ) = 1 at all cusps κ and so that all possible intersections on X( K ) of T (p), div (F ), div(G) are proper. Such Borcherds products exist by Theorem 4.12. We take the pairs (div N (F ), g N (F )) and (div N (G), g N (G)) as representatives for c 1 M k ( K (N)) , where we have put g N (F ) = (2πikω, − log F σ (z 1 , z 2 ) ) σ ∈ and g N (G) analogously. With our convention on arithmetic intersection numbers, we have (6.5) For the first summand of (6.5), we obtain, by Proposition 5.15, deg (h N ) * (T N (p) · div N (F ) · div N (G)) = deg(π N ) · deg g * (div(ϕ * F ) · div(ϕ * G)) + γ p,k log(p) for some γ p,k ∈ Q.
Since F and G are integral Borcherds products, they are invariant under Aut(C/Q). Therefore, by means of Remark 3.5, we find, for the second summand of (6.5), K ) g(p) * g(F ) * g (G).
Here, A N (τ ) is the arithmetic generating series (6.3), and E(τ ) is the holomorphic Eisenstein series of weight 2 (see (4.2)). In particular, for the arithmetic self-intersection number of M k ( K (N)), we have, in R N , (6.6)

Proof
Throughout the proof, all equalities that contain the image of the arithmetic degree map deg are equalities in R N .
with χ D (p) = 1. Therefore In the last equality, we have used the relation a(m) vol(T (m)) = (ζ K (−1)/2)k − p a(p) vol(T (p)). Hence the claim follows from (4.3). (6.7) The first term is computed in Theorem 6.4. The integral is, by functoriality of the intersection product, equal to d N times the quantity from (3.18) of Corollary 3.9. ᮀ Remark 6.6 (i) Observe that in view of Lemma 3.1, we may calculate formula (6.7) for all T (m) and, in particular, also for isotropic Hirzebruch-Zagier divisors. We may take this quantity as an ad hoc definition for the Faltings height of T N (m) with respect to M k ( K (N)). For example, if m is square free and χ D (p) = 1 for all primes p dividing m, then up to boundary components, the normalization of T N (m) is isomorphic to some modular curve (see Remark 2.16). Although T N (m) / ∈ Z 1 U ( H(N)), we obtain, in R N , Up to the sum over primes dividing the level m, this equals the arithmetic selfintersection number of the line bundle of modular forms on that modular curve as computed in [Kü2]. It would be interesting to obtain a geometric interpretation of the difference using the change of moduli problems over Z extending Proposition 5.11. At the primes dividing m, the morphism of line bundles (5.6) is not necessarily an isomorphism anymore.
(ii) Recall that if T (m) ⊂ X( K ) is anisotropic, then its normalization is isomorphic to some Shimura curve of discriminant m. For example, if m is square free with an even number of prime factors, and χ D (p) = −1 for all primes p dividing m, then T (m) is anisotropic and our formula gives, in R N , (6.8) Related formulas have been obtained by Maillot and Roessler (see [MR1,Proposition 2.3]) using completely different techniques.
In recent work, Kudla, Rapoport, and Yang stated a conjecture for the arithmetic self-intersection number of the Hodge bundle on a Shimura curve (see [KRY,(0.17),p. 891]). In the forthcoming work [KK], it will be shown that (6.8), together with versions of Propositions 3.11 and 5.11 for Shimura curves, can be used to obtain a proof of their conjecture. where the first map is the diagonal and the second is the difference, is exact. Thus we have an isomorphism lim ← −N≥3 R N ∼ = R. ᮀ By abuse of notation, given an element x ∈ CH 3 ( H, D pre ), we denote also by x its image deg(x) ∈ R under the arithmetic degree map.

R-valued arithmetic intersection theory
Then Theorem 6.4 implies the following. THEOREM 6.8 We have the following identity of modular forms: Analogously, we can define a height pairing

Remark 6.11
Finally, we note that if there were an arithmetic Chow ring for the stack H(1) satisfying the usual functorialities, then it would map to CH * ( H, D pre ), and the arithmetic degrees, as well as the values of the height pairing, would coincide.
One can also follow different strategies to obtain R-valued arithmetic intersection numbers. First, we note that the proofs of Theorems 6.8 and 6.9 would directly carry over to other level structures. For instance, Pappas constructed regular models over Z of Hilbert modular surfaces associated to congruence subgroups 00 (A) ⊂ K (see [P]). If there exists a toroidal compactification over Z of H Fil 00 (A) (see [P, p. 51]), one can work on that arithmetic variety and derive formulas as in Theorems B and C. Alternatively, one can try to work directly on the coarse moduli space. This requires arithmetic Chow groups for singular arithmetic varieties of some kind (e.g., for arithmetic Baily-Borel compactifications of Hilbert modular surfaces), equipped with arithmetic Chern-class operations for hermitian line bundles.