Evaluating the wild Brauer group

Abstract

Classifying elements of the Brauer group of a variety X over a p-adic field by the p-adic accuracy needed to evaluate them gives a filtration on operatorname{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.