Wild Brauer classes via prismatic cohomology
Emiliano Ambrosi, Rachel Newton, Margherita Pagano
TL;DR
The article addresses the problem of producing p-torsion Brauer classes on smooth proper varieties over p-adic fields with good reduction when a nonvanishing global 2-form appears on the generic fibre. By combining a cohomological perspective with a prismatic-cohomology framework, the authors establish a Newton–above–Hodge theorem for mod p prismatic cohomology and relate p-adic vanishing cycles to Brauer-class data, enabling explicit constructions in special cases such as products of elliptic curves and abelian varieties. The main result shows that under a mild Hodge-number equality, the existence of a global 2-form forces p-torsion Brauer classes with surjective evaluation maps after finite extensions, yielding transcendental obstructions to weak approximation in arithmetic settings. This provides new global implications: varieties over number fields that satisfy weak approximation for all finite extensions must have vanishing extremal Hodge numbers, i.e., $H^0(Z,\Omega^2)=0$, and clarifies the prevalence of potentially relevant Brauer obstructions at almost all places. Methodologically, the work develops a robust bridge between Brauer residues, henselisation, p-adic vanishing cycles, and prismatic cohomology, yielding concrete, computable descriptions in several important families (notably products of elliptic curves and Kummer varieties). The results generalize prior ordinary-reduction phenomena, offering broad new instances of transcendental Brauer classes obstructing weak approximation.
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.