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- W2952805966 abstract "Let $A/mathbb{Q}$ be an abelian variety of dimension $ggeq 1$ 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." @default.
- W2952805966 created "2019-06-27" @default.
- W2952805966 creator A5042240850 @default.
- W2952805966 creator A5079532710 @default.
- W2952805966 date "2015-11-07" @default.
- W2952805966 modified "2023-09-27" @default.
- W2952805966 title "Fields of definition of elliptic $k$-curves and the realizability of all genus 2 Sato--Tate groups over a number field" @default.
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