Uniform boundedness for Brauer groups of forms in positive characteristic

Let $k$ be a finitely generated field of characteristic $p \gt 0$ 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 $Br(X_{\overline{k}}) {[p^\prime]}^{\pi_1(k)}$ of the prime-to-$p$ torsion of the geometric Brauer group of $X$ is finite. The main result of this note is that, if $X$ satisfies the $\ell$-adic Tate conjecture for divisors for every prime $\ell \neq p$, for every integer $d \geq 1$, there exists a constant $C := C(X, d)$ such that for every finite field extension $k \subseteq k^\prime$ with ${[k^\prime : k]} \leq d$ and every $(\overline{k} / k^\prime)$-form $Y$ of $X$ one has $\lvert (Br(Y \times_{k^\prime} \overline{k}) {[p^\prime]}^{\pi_1 (k^\prime)} \rvert \leq C$. 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.

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Received 26 June 2019

Accepted 17 July 2019

Published 13 May 2021