Please use this identifier to cite or link to this item: http://hdl.handle.net/2445/142922
 Title: Fields of definition of elliptic k-curves and the realizability of all genus 2 Sato-Tate groups of over a number field Author: Fité i Llevot, Francesc, 1948-Guitart Morales, Xavier Keywords: Corbes el·líptiquesTeoria de grupsElliptic curvesGroup theory Issue Date: 18-Jan-2018 Publisher: American Mathematical Society (AMS) Abstract: Let $A/\mathbb{Q}$ be an abelian variety of dimension $g\geq 1$ that is isogenous over $\overline {\mathbb{Q}}$ to $E^g$, where $E$ is an elliptic curve. If $E$ does not have complex multiplication (CM), by results of Ribet and Elkies concerning fields of definition of elliptic $\mathbb{Q}$-curves, $E$ is isogenous to a curve defined over a polyquadratic extension of $\mathbb{Q}$. We show that one can adapt Ribet's methods to study the field of definition of $E$ up to isogeny also in the CM case. We find two applications of this analysis to the theory of Sato-Tate groups: First, we show that $18$ of the $34$ possible Sato-Tate groups of abelian surfaces over $\mathbb{Q}$ occur among at most $51$ $\overline {\mathbb{Q}}$-isogeny classes of abelian surfaces over $\mathbb{Q}$. Second, we give a positive answer to a question of Serre concerning the existence of a number field over which abelian surfaces can be found realizing each of the $52$ possible Sato-Tate groups of abelian surfaces. Note: Versió postprint del document publicat a: https://doi.org/10.1090/tran/7074 It is part of: Transactions of the American Mathematical Society, 2018, vol. 370, num. 7, p. 4623-4659 URI: http://hdl.handle.net/2445/142922 Related resource: https://doi.org/10.1090/tran/7074 ISSN: 0002-9947 Appears in Collections: Articles publicats en revistes (Matemàtiques i Informàtica)

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