Lattices in Tate modules

TL;DR

This paper refines Zarhin's theorem by showing that for a g-dimensional abelian variety and an endomorphism, there exists an integral matrix representing the endomorphism on Tate modules and Dieudonné modules, providing a concrete linear algebraic description.

Contribution

The paper proves the existence of a single integral matrix representing endomorphisms on Tate and Dieudonné modules, refining previous results by Zarhin.

Findings

Existence of a matrix A in M_{2g}(Z) representing the endomorphism u on Tate modules.

Existence of a matrix A representing u on Dieudonné modules over perfect fields of characteristic p.

Provides a basis on which the endomorphism acts via a fixed integral matrix.

Abstract

Refining a theorem of Zarhin, we prove that given a $g$-dimensional abelian variety $X$ and an endomorphism $u$ of $X$, there exists a matrix $A \in \operatorname{M}_{2g}(\mathbb{Z})$ such that each Tate module $T_\ell X$ has a $\mathbb{Z}_\ell$-basis on which the action of $u$ is given by $A$, and similarly for the covariant Dieudonn\'e module tensored with $\mathbb{Q}$ if over a perfect field of characteristic $p$.