On the Tate and Standard Conjectures over Finite Fields

TL;DR

This paper extends results relating to the Tate and Standard Conjectures from abelian varieties to motives and other varieties like K3 surfaces, providing new criteria and generalizations.

Contribution

It generalizes previous results on Tate and Standard Conjectures to motives and applies them to a broader class of varieties including K3 surfaces.

Findings

Established conditions under which l-homological and numerical equivalence coincide for motives.

Provided criteria linking Tate's theorem on divisors to the Tate conjecture.

Extended the applicability of these conjectures to K3 surfaces and other varieties.

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.