The Tate and Standard Conjectures for Certain Abelian Varieties

TL;DR

This paper demonstrates that the Tate and standard conjectures hold for many abelian varieties over finite fields, including some with complex algebraic structures, by refining previous proofs to apply to individual varieties.

Contribution

It refines earlier proofs to establish the Tate and standard conjectures for specific abelian varieties over finite fields, expanding their applicability.

Findings

Proves Tate and standard conjectures for many abelian varieties over finite fields.

Includes varieties with Tate classes not generated by divisor classes.

Provides unconditional proofs for certain classes of abelian varieties.

Abstract

In two earlier articles, we proved that, if the Hodge conjecture is true for ALL CM abelian varieties over the complex numbers, then both the Tate conjecture and the standard conjectures are true for abelian varieties over finite fields. Here we rework the proofs so that they apply to a single abelian variety. As a consequence, we prove (unconditionally) that the Tate and standard conjectures are true for many abelian varieties over finite fields, including abelian varieties for which the algebra of Tate classes is not generated by divisor classes.