Mathematical Research Letters

Volume 14, Issue 1, January 2007 pp. 157-164.

Abelian varieties without homotheties

Authors: Yuri G. Zarhin
Author institution: Pennsylvania State University

Summary: 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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