Let $X$ be a (smooth projective geometrically connected) surface over a finitely generated field $k$. The Tate conjecture predicts that, for $l$ a prime number invertible in $k$, the Chern class map from Pic$(X) \otimes \mathbb Q\_l$ to the Galois invariants of the second etale cohomology group of $X$ is surjective (and thus bijective).


This conjecture is known if X is dominated by a product of curves.


Is it known if X is an etale cover of a product of curves? (We can restrict our attention to the case that X is of positive Kodaira dimension.)


Etale covers of a product of curves are not necessarily a product of curves. We would be done if etale covers of a product of curves are dominated by a product of curves. Is that the case?



Is the Tate conjecture known for etale covers of products of curves
The answer to the last question (and therefore to the others) is yes. An étale cover of a variety $X$ is dominated by an étale Galois $G$-cover, for some finite group $G$; and this is given by a homomorphism $u:\pi \_1(X)\rightarrow G$. If $X=X\_1\times \ldots \times X\_p$, this gives homomorphisms $u\_i:\pi \_1(X\_i)\rightarrow G$, and there are étale coverings $\tilde{X}\_i\rightarrow X\_i $ such that the composition $\pi \_1(\tilde{X}\_i )\rightarrow \pi \_1(X\_i)\rightarrow G$ is trivial. This implies that your étale cover is dominated by $\tilde{X}\_1\times \ldots \times \tilde{X}\_p $.


