Lattices in Tate modules.

Poonen, Bjorn; Rybakov, Sergey · Proc Natl Acad Sci U S A · 2021

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Abstract

Refining a theorem of Zarhin, we prove that, given a <i>g</i>-dimensional abelian variety <i>X</i> and an endomorphism <i>u</i> of <i>X</i>, 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 <i>u</i> is given by <i>A</i>, and similarly for the covariant Dieudonné module if over a perfect field of characteristic <i>p</i>.