Endomorphism fields of abelian varieties

TL;DR

This paper establishes a precise divisibility bound on the degree of the field extension needed to realize endomorphisms of an abelian variety, refining previous results and connecting to the structure of the Sato-Tate group.

Contribution

It provides a sharper bound on the field extension degree for endomorphisms of abelian varieties, extending prior work and applying Minkowski's reduction method in complex cases.

Findings

Derived a divisibility bound in terms of g for endomorphism fields.

Connected the bound to the order of the Sato-Tate group component.

Extended results previously known for abelian surfaces to higher dimensions.

Abstract

We give a sharp divisibility bound, in terms of g, for the degree of the field extension required to realize the endomorphisms of an abelian variety of dimension g over an arbitrary number field; this refines a result of Silverberg. This follows from a stronger result giving the same bound for the order of the component group of the Sato-Tate group of the abelian variety, which had been proved for abelian surfaces by Fite-Kedlaya-Rotger-Sutherland. The proof uses Minkowski's reduction method, but with some care required in the extremal cases when p equals 2 or a Fermat prime.