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

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