Evaluating the wild Brauer group

The Brauer group of a variety over a number field is a powerful tool for studying failures of the Hasse principle. To apply this, one needs to understand the function obtained by evaluating a given element of the Brauer group at the p-adic points of the variety. For elements of order coprime to p, this evaluation map factors through reduction modulo p. For elements of order p, the situation is much more intricate: one may need to look at the points to higher p-adic precision. We relate the resulting filtration on the Brauer group to one defined by Bloch–Kato, and show that Kato's refined Swan conductor controls the local behaviour of the evaluation map. This is joint work with Rachel Newton.