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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
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Let A/ℚ be an abelian variety of dimension g≥1 that is isogenous over ℚ to Eᵍ, 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 ℚ-curves E is isogenous to a curve defined over a polyquadratic extension of 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 ℚ occur among at most $51$ ℚ-isogeny classes of abelian surfaces over 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.
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