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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?

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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 $.

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    $\begingroup$ The key point is that the fundamental group of a product is the product of the fundamental groups. This is only true for complete varieties, but given the context of the Tate conjecture the varieties are complete. $\endgroup$ – Ben Wieland Jul 23 '14 at 16:27
  • $\begingroup$ Right. I had in mind the case where each $X_i$ is a smooth projective curve, as in the question. $\endgroup$ – abx Jul 23 '14 at 16:52

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