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

Francesc Fité, Xavier Guitart

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 comple…