Standard conjecture of Hodge type for powers of abelian varieties

TL;DR

This paper proves the standard conjecture of Hodge type for powers of abelian threefolds and related cases over finite fields, using tannakian categories and Frobenius eigenvalues.

Contribution

It establishes the conjecture for powers of abelian threefolds and simple abelian varieties of prime dimension over finite fields, introducing new methods involving fiber functors and Frobenius eigenvalues.

Findings

Proved the conjecture for abelian threefolds' powers.

Extended results to simple abelian varieties of prime dimension.

Developed explicit descriptions of Lefschetz motives over finite fields.

Abstract

We prove that the standard conjecture of Hodge type holds for powers of abelian threefolds. Along the way, we also prove the conjecture for powers of simple abelian variety of prime dimension over finite fields, and in other related cases based on the notion of Frobenius rank of Lenstra-Zarhin. The main tool is a result comparing two real fiber functors on tannakian categories. A second tool is a new an explicit description of simple Lefschetz motives over finite fields, in terms of ''enriched'' Frobenius eigenvalues.