Rational Tate сlasses

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.