Sur la conjecture de Tate enti\`ere pour le produit d'une courbe et d'une surface $CH_{0}$-triviale sur un corps fini

TL;DR

This paper proves a strong version of the integral Tate conjecture for 1-cycles on the product of a curve and a surface over a finite field, assuming the surface is geometrically $CH_0$-trivial, with applications to Enriques surfaces.

Contribution

It establishes the conjecture for products involving geometrically $CH_0$-trivial surfaces, extending previous results and analyzing unramified cohomology groups.

Findings

Proves the integral Tate conjecture for certain products over finite fields.

Applies to Enriques surfaces with geometrically $CH_0$-triviality.

Analyzes unramified cohomology to support the conjecture.

Abstract

We investigate a strong version of the integral Tate conjecture for 1-cycles on the product of a curve and a surface over a finite field, under the assumption that the surface is geometrically $CH_0$-trivial. By this we mean that over any algebraically closed field extension, the degree map on the zero-dimensional Chow group of the surface is an isomorphism. This applies to Enriques surfaces. When the N\'eron-Severi group has no torsion, we recover earlier results of A. Pirutka. The results rely on a detailed study of the third unramified cohomology group of specific products of varieties.