  In despair, as Deligne (2000) put it, of proving the Hodge and Tate
conjectures, we can try to find substitutes. For abelian varieties in
characteristic zero, Deligne (1982) 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. v3. Submitted version.
