Uniform boundedness for Brauer groups of forms in positive characteristic (Q2028608)

The object of study in the article under review is the (geometric) Brauer group \(\mathrm{Br}(X_{\overline{k}}) = \mathrm{H}^2 \left( X_{\overline{k}}, \mathbb{G}_m\right)\) of a smooth proper scheme \(X\) over a finitely generated field \(k\) of positive characteristic \(p\) with algebraic closure \(\overline{k}\). For a prime number \(\ell\) different from \(p\), the \(\ell\)-adic Tate conjecture on divisors predicts that the \(\ell\)-adic cycle class map \[c_1: \mathrm{Pic}\left( X_{\overline{k}} \right)\otimes \mathbb{Q}_\ell \rightarrow \bigcup_{[k':k] \text{ finite}} \mathrm{H}^2 \left( X_{\overline{k}}, \mathbb{Q}_\ell(1)\right)^{\pi_1(k')}\] is surjective, where \(\pi_1(k')\) denotes the absolute Galois group of \(k'\) [\textit{J. Tate}, Proc. Conf. Purdue Univ. 1963, 93--110 (1965; Zbl 0213.22804)]. This conjecture is equivalent to the finiteness of the \(\ell\)-primary torsion of \(\mathrm{Br}\left( X_{\overline{k}}\right)^{\pi_1(k')}\) for every finite extension \(k'\) of \(k\). It has been shown that abelian varieties and \(K3\) surfaces satisfy the \(\ell\)-adic Tate conjecture on divisors (see e.g. [\textit{J. Tate}, Proc. Symp. Pure Math. 55, 71--83 (1994; Zbl 0814.14009)]). As a strengthening of this equivalence, in [\textit{A. Cadoret} et al., ``\(\mathbb{Q}_\ell\)-versus \(\mathbb{F}_\ell\)-coefficients in the Grothendieck-Serre-Tate conjectures'', Preprint], the authors show that, if \(X\) satisfies the \(\ell\)-adic Tate conjecture for divisors for every \(\ell\), then the prime-to-\(p\) torsion \( \left( \mathrm{Br} \left( X_{\overline{k}} \right)[p']\right)^{\pi_1(k)}\) is finite. However, it is difficult to determine explicit bounds. In the paper under review, it is shown that \( \mathrm{Br} \left( X_k \right)[p']^{\pi_1(k)}\) is uniformly bounded. In particular, for every integer \(d \geq 1\), there exists a constant \(C =C(X,d)\) that satisfies the following: \\ Let \(k \subseteq k' \subseteq \overline{k}\) be a field extension with \([k':k] \leq d\) and let \(Y\) be a \(\left(\overline{k}/k'\right)\)-form of \(X\), then \[ \left|\left( \mathrm{Br} \left( Y_{\overline{k}} \right)[p']\right)^{\pi_1(k')}\right| \leq C.\] An equivalent result was shown in characteristic zero in [\textit{M. Orr} and \textit{A. N. Skorobogatov}, Compos. Math. 154, No. 8, 1571--1592 (2018; Zbl 1452.14016)]. The main ingredients in the proof are two results on compatible systems of \(\pi_1(k)\) representations (Proposition 1.2.2.1 and Proposition 1.2.2.2).