Fields of definition of elliptic k-curves and the realizability of all genus 2 Sato-Tate groups of 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.