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 $g\geq 1$ 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.