Wild Brauer classes via prismatic cohomology

Abstract

Let $K$ be a finite extension of $\mathbb{Q}_p$ and $X$ a smooth proper $K$-variety with good reduction. Under a mild assumption on the behaviour of Hodge numbers under reduction modulo $p$, we prove that the existence of a non-zero global 2-form on $X$ implies, after a finite extension of $K$, the existence of $p$-torsion Brauer classes with surjective evaluation map. This implies that any smooth proper variety over a number field which satisfies weak approximation over all finite extensions has no non-zero global 2-form. The proof is based on a prismatic interpretation of Brauer classes with eventually constant evaluation, and a Newton-above-Hodge result for the mod $p$ reduction of prismatic cohomology. This generalises work of Bright and the second-named author beyond the ordinary reduction case.