let $a$ be an abelian variety defined over a number field $k$ that is isogenous over an algebraic closure to the power of an elliptic curve $e$. if $e$ does not have cm, by results of ribet and elkies concerning fields of definition of $k$-curves, $e$ is isogenous to an elliptic curve defined over a polyquadratic extension of $k$. 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 of abelian surfaces: first, we show that 18 of the 34 possible sato-tate groups of abelian surfaces over $\mathbb{q}$, only 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. this is a joint work with xevi guitart. preparatory talk: in the preparatory talk i plan to review very briefly basic definitions concerning abelian varieties necessary to introduce (in the main talk) the notion of abelian $k$-variety. i will also present the (general) sato-tate conjecture and show how it motivates the problem considered in the main talk.