On the Tate and Standard Conjectures over Finite Fields

Abstract

For an abelian variety over a finite field, Clozel (1999) showed that l-homological equivalence coincides with numerical equivalence for infinitely many l, and the author (1999) gave a criterion for the Tate conjecture to follow from Tate's theorem on divisors. We generalize both statements to motives, and apply them to other varieties including K3 surfaces.