On matrices of endomorphisms of abelian varieties

TL;DR

This paper investigates the structure of endomorphisms of abelian varieties by analyzing their action on l-adic Tate modules, establishing a basis where the matrices have rational entries independent of l.

Contribution

It proves the existence of a basis for Tate modules where endomorphism matrices have rational entries that are consistent across all l.

Findings

Matrices of endomorphisms can be represented with rational entries independent of l.

A basis exists for Tate modules that simplifies the study of endomorphisms.

The approach unifies the understanding of endomorphisms across different primes l.

Abstract

We study endomorphisms of abelian varieties and their action on the l-adic Tate modules. We prove that for every endomorphism one may choose a basis of each Tate module such that the corresponding matrix has rational entries and does not depend on l.