Wild Brauer classes via prismatic cohomology

2025/09/26 by Ambrosi, Emiliano, Newton, Rachel, Pagano, Margherita
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2509.22025

Abstract

Let K be a finite extension of 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.

Citations

• The role of primes of good reduction in the Brauer--Manin obstruction
• Logarithmic prismatic cohomology II
• Absolute prismatic cohomology
• Essential dimension via prismatic cohomology
• Prismatic Dieudonné theory
• Prisms and Prismatic Cohomology
• Odd order obstructions to the Hasse principle on general K3 surfaces
• Topological Hochschild homology and integral p-adic Hodge theory
• p-adic vanishing cycles as Frobenius-fixed points
• Kummer varieties and their Brauer groups
• Odd order Brauer--Manin obstruction on diagonal quartic surfaces
• The Brauer group of Kummer surfaces and torsion of elliptic curves
• Diagonal quartic surfaces and transcendental elements of the Brauer group

Related

• Prismatic Steenrod operations and arithmetic duality on Brauer groups
• On p-torsions of geometric Brauer groups
• Kummer varieties and their Brauer groups
• Splitting unramified Brauer classes by abelian torsors and the period-index problem
• Remarks on p-primary torsion of the Brauer group