This article is cited in 2 papers

Rational Tate сlasses

J. S. Milne

Mathematics Department, University of Michigan, Ann Arbor, MI, USA

Abstract: In despair, as Deligne put it, of proving the Hodge and Tate conjectures, one can try to find substitutes. For abelian varieties in characteristic zero, Deligne in his 1978–1979 IHES seminar constructed a theory of Hodge classes having many of the properties that the algebraic classes would have if the Hodge conjecture were known. In this article I investigate whether there exists a theory of “rational Tate classes” on varieties over finite fields having the properties that the algebraic classes would have if the Hodge and Tate conjectures were known. In particular, I prove that there exists at most one “good” such theory.

Key words and phrases: abelian varieties, finite fields, Tate conjecture.

MSC: 14C25, 14K15, 11G10

Received: April 30, 2008

Language: English

DOI: 10.17323/1609-4514-2009-9-1-111-141

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