Lattices in Tate modules

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 $A\in M_{2g}(\mathbb{Z})$ 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 Dieudonné module tensored with $\mathbb{Q}$ if over a perfect field of characteristic $p$.

4 pages. This version includes a statement for Dieudonné modules as well as Tate modules, and corrects an error in the published version