Lattices in Tate modules

TL;DR

The paper addresses the problem of representing the action of a fixed endomorphism $u$ of an abelian variety $X$ uniformly on all $\ell$-adic Tate modules and the covariant Dieudonné module by a single integral matrix. It builds on Zarhin's result by proving the existence of $u$-stable lattices in the adelic and $p$-adic realizations, yielding a common matrix $A \in \mathrm{M}_{2g}(\mathbb{Z})$ that describes $u$ across all realizations. The proof reduces to an isotypic decomposition via the endomorphism algebra, uses freeness over number-field tensor products, and employs local lattice theory and Ribet-type arguments to pass from rational to integral lattices in both the equal-characteristic and $p$-adic settings. The work also analyzes generalizations to $R$-stable lattices, showing positive results when $R$ sits inside a semisimple matrix-algebra decomposition, but presenting counterexamples that arise when $R$ is commutative in the presence of quaternionic endomorphisms, thereby clarifying the limitations of a uniform integral lattice theory.

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 $p$.

Theorems & Definitions (9)

• Definition 1.1
• Theorem 1.2
• Remark 1.3
• Corollary 1.4
• Lemma 2.1
• proof
• Lemma 2.2
• proof
• proof : Proof of Theorem \ref{['T:main']}