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. 2021 Dec 7;118(49):e2113201118.
doi: 10.1073/pnas.2113201118.

Lattices in Tate modules

Bjorn Poonen  1 , Sergey Rybakov  2   3
Affiliations

Lattices in Tate modules

Bjorn Poonen et al. Proc Natl Acad Sci U S A. .

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 [Formula: see text] such that each Tate module [Formula: see text] has a [Formula: see text]-basis on which the action of u is given by A, and similarly for the covariant Dieudonné module if over a perfect field of characteristic p.

Keywords: Dieudonné module; Tate module; abelian variety; endomorphism.

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Conflict of interest statement

The authors declare no competing interest.

References

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    1. Chai C. L., Conrad B., Oort F., Complex Multiplication and Lifting Problems (Mathematical Surveys and Monographs, American Mathematical Society, Providence, RI, 2014), vol. 195.
    1. Demazure M., Lectures on p-Divisible Groups (Lecture Notes in Mathematics, Springer, 1972), vol. 302.

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