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Abelian varieties without homotheties

Yuri G. Zarhin
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Abstract

A celebrated theorem of Bogomolov asserts that the $\ell$-adic Lie algebra attached to the Galois action on the Tate module of an abelian variety over a number field contains all homotheties. This is not the case in characteristic $p$: a "counterexample" is provided by an ordinary elliptic curve defined over a finite field. In this note we discuss (and explicitly construct) more interesting examples of "non-constant" absolutely simple abelian varieties (without homotheties) over global fields in characteristic $p$.

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Yuri G. Zarhin. 2006-10-16. Abelian varieties without homotheties. https://arxiv.org/abs/math/0606422

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