Uniform boundedness for Brauer groups of forms in positive characteristic

Abstract

Let $k$ be a finitely generated field of characteristic $p>0$ and $X$ a smooth and proper scheme over $k$. Recent works of Cadoret, Hui and Tamagawa show that, if $X$ satisfies the $\ell$-adic Tate conjecture for divisors for every prime $\ell \ne p$, the Galois invariant subgroup $Br(X_{\overline{k}})[p^{\prime }{]}^{{\pi }_1(k)}$ of the prime-to-$p$ torsion of the geometric Brauer group of $X$ is finite. The main result of this note is that, for every integer $d\ge 1$, there exists a constant $C:=C(X,d)$ such that for every finite field extension $k\subseteq k^{\prime }$ with $[k^{\prime }:k]\le d$ and every $(\overline{k}/k^{\prime })$-form $Y$ of $X$ one has $|(Br(Y{\times }_{k^{\prime }}\overline{k})[p^{\prime }{]}^{{\pi }_1(k^{\prime })}|\le C$. The theorem is a consequence of general results on forms of compatible systems of ${\pi }_1(k)$-representations and it extends to positive characteristic a recent result of Orr and Skorobogatov in characteristic zero.

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