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 $\ell\neq p$, the Galois invariant subgroup $Br(X_{\overline k})[p']^{\pi_1(k)}$ 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 $d\geq 1$, there exists a constant $C:=C(X,d)$ such that for every finite field extension $k \subseteq k'$ with $[k':k]\leq d$ and every $(\overline k/k')$-form $Y$ of $X$ one has $|(Br(Y\times_{k'}\overline k)[p']^{\pi_1(k')}|\leq C$. 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.