Evaluating the wild Brauer group

Abstract

Classifying elements of the Brauer group of a variety X over a p-adic field according to the p-adic accuracy needed to evaluate them gives a filtration on Br X. We relate this filtration to that defined by Kato's Swan conductor. The refined Swan conductor controls how the evaluation maps vary on p-adic discs: this provides a geometric characterisation of the refined Swan conductor. We give applications to rational points on varieties over number fields, including failure of weak approximation for varieties admitting a non-zero global 2-form.

58 pages; minor changes. Final version. The Version of Record of this article is published in Inventiones Mathematicae and is available online at https://doi.org/10.1007/s00222-023-01210-8

References in corpus (2)

• Coincidence of two Swan conductors of abelian characters
• A characterization of ramification groups of local fields with imperfect residue field

Cited by in corpus (2)

• An example of a Brauer-Manin obstruction to weak approximation at a prime with good reduction
• Transcendental Brauer-Manin obstructions on singular K3 surfaces