Reduction of Brauer classes on K3 surfaces, rationality and derived equivalence (Q2093203)

The aim of the paper under review is to study the reductions of the Brauer class for a polarized \(K3\) surface over a number field \(k\). In particular, it is shown as the main theorem that the specialization of it at a place of good reduction of positive natural density is trivial. The idea for the proof is to use the result (\(l\)-adic open image theorems) of integral version of the Mumford-Tate conjecture. Besides, they give two applications. The first is that for a certain cubic \(4\)-fold \(Y\), there exists a set of places \(\mathfrak{p}\) of \(k\) of positive natural density for which the reduction \(Y_{\mathfrak{p}}\) at \(\mathfrak{p}\) is rational. The second is that for a \(K3\) surface \(X\) of degree \(2d\) over \(k\) and an associated twisted \(K3\) surface \((M,\,\alpha)\) which parametrizes geometrically stable sheaves on \(X\), there exists a set of places \(\mathfrak{p}\) of \(k\) of positive natural density such that the \(\mathfrak{p}\)-reduction \(M_{\mathfrak{p}}\) is a fine moduli space, and there obtains an \(\mathbb{F}_{\mathfrak{p}}\)-linear derived equivalence between the categories \(D^b(X_{\mathfrak{p}})\) and \(D^b(M_{\mathfrak{p}})\).