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

TL;DR

This paper investigates the fields of definition of elliptic $k$-curves, extending Ribet's methods to CM cases, and applies these results to classify Sato--Tate groups of abelian surfaces over number fields.

Contribution

It extends Ribet's techniques to CM elliptic curves and demonstrates that most Sato--Tate groups of abelian surfaces occur over a limited number of isogeny classes, also confirming their realizability over some number field.

Findings

18 of 34 Sato--Tate groups occur among at most 51 isogeny classes

All 52 Sato--Tate groups can be realized over some number field

Extension of Ribet's methods to CM elliptic curves

Abstract

Let $A/\mathbb{Q}$ be an abelian variety of dimension $g\geq 1$ 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.