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Which information is currently known about $H^1_{et}(X,\mathbb{Z}_l)$ and $H^2_{et}(X,\mathbb{Z}_l)$, where $X$ is a smooth unirational variety over an algebraically closed field of finite characterstic $p\neq l$? Actually, I am interested in the case where $X$ is a quotient of a rational variety $Y$ by a free action of a finite group (whose order is prime to $p$). Does the situation differ much from the complex variety case; are there any examples where it does?

Certainly, my question is closely related to (the torsion) of $\operatorname{Pic}(X)$ and to the Brauer group of $X$.

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    $\begingroup$ If you want your varieties to behave like in characteristic $0$, it's probably better to say separably unirational. In positive characteristic there are all sorts of 'strange' unirational varieties; for example there exist unirational K3 surfaces. Maybe this doesn't matter for your question, but it seems that you're thinking about the separable case anyway. $\endgroup$ Feb 18, 2020 at 2:10

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Concerning the $H^2$, for $X$ a smooth projective rationally chain connected variety over an algebraically closed field $k$ and $\ell \in k^\ast$, it follows from Theorem 1.2 in https://arxiv.org/abs/1703.05735 that

$$\mathrm{NS}(X)\otimes \mathbb{Z}_{\ell} = \mathrm{H}^2_{et}(X,\mathbb{Z}_{\ell}(1))$$

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