Abelian varieties without homotheties

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A celebrated theorem of Bogomolov asserts that the ℓ-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.

Original languageEnglish (US)
Pages (from-to)157-164
Number of pages8
JournalMathematical Research Letters
Issue number1
StatePublished - Jan 2007

All Science Journal Classification (ASJC) codes

  • General Mathematics


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