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arXiv · 2107.06363

Lattices in Tate modules

Bjorn Poonen·Sergey Rybakov
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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 Dieudonné module tensored with $\mathbb{Q}$ if over a perfect field of characteristic $p$.

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Bjorn Poonen, Sergey Rybakov. 2025-10-14. Lattices in Tate modules. https://doi.org/10.1073/pnas.2113201118

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