Fields of definition of elliptic $k$-curves and the realizability of all genus 2 Sato--Tate groups over a number field

Abstract

Let $A/\mathbb{Q}$ be an abelian variety of dimension $g\ge 1$ 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.

Erratum added at the end of the Introduction summarizing some small changes made on the published version in Trans. Amer. Math. Soc. None of the changes affects the main results of the paper

References in corpus (2)

• Sato-Tate distributions and Galois endomorphism modules in genus 2
• Modular abelian varieties over number fields