Evaluating the wild Brauer group

TL;DR

This paper investigates the structure of the Brauer group over p-adic fields, relating it to Swan conductors, and explores implications for rational points and weak approximation on varieties over number fields.

Contribution

It introduces a new filtration on the Brauer group based on p-adic evaluation accuracy and links it to Kato's Swan conductor, providing geometric insights.

Findings

Relation between the Brauer group filtration and Swan conductor.

Geometric characterization of the refined Swan conductor.

Applications to failures of weak approximation on certain varieties.

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.