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Let $A/\mathbb{Q}$ be an abelian variety of dimension $g\geq 1$ that isisogenous over $\overline{\mathbb{Q}}$ to $E^g$, where $E$ is an ellipticcurve. If $E$ does not have complex multiplication (CM), by results of Ribetand 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 thisanalysis to the theory of Sato--Tate groups: First, we show that $18$ of the$34$ possible Sato--Tate groups of abelian surfaces over $\mathbb{Q}$ occuramong at most $51$ $\overline{\mathbb{Q}}$-isogeny classes of abelian surfacesover $\mathbb{Q}$; Second, we give a positive answer to a question of Serreconcerning the existence of a number field over which abelian surfaces can befound realizing each of the $52$ possible Sato--Tate groups of abeliansurfaces.

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