Autour de la conjecture de Tate enti\`ere pour certains produits de dimension $3$ sur un corps fini

TL;DR

This paper investigates a strong form of the integral Tate conjecture for 1-cycles on products of a surface and a curve over a finite field, extending previous results with unconditional proofs.

Contribution

It generalizes and provides unconditional proofs for several results related to the integral Tate conjecture for specific algebraic varieties.

Findings

Established new unconditional results for the integral Tate conjecture.

Extended previous work to broader classes of products of surfaces and curves.

Contributed to the understanding of algebraic cycles over finite fields.

Abstract

Let $X$ be the product of a surface satisfying $b_2=\rho$ and of a curve over a finite field. We study a strong form of the integral Tate conjecture for $1$-cycles on $X$. We generalize and give unconditional proofs of several results of our previous paper with J.-L. Colliot-Th\'el\`ene.