  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 $A$.
