Mathematical Research Letters

Volume 14 (2007)

Number 1

Abelian varieties without homotheties

Pages: 157 – 164

DOI: http://dx.doi.org/10.4310/MRL.2007.v14.n1.a13

Author

Yuri G. Zarhin (Pennsylvania State University)

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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