Evaluating the wild Brauer group

Abstract: The local-global approach to the study of rational points on varieties over number fields begins by embedding the set of rational points on a variety $X$ into the set of its adelic points. The Brauer–Manin pairing cuts out a subset of the adelic points, called the Brauer–Manin set, that contains the rational points. If the set of adelic points is non-empty but the Brauer–Manin set is empty then we say there's a Brauer–Manin obstruction to the existence of rational points on $X$. Computing the Brauer–Manin pairing involves evaluating elements of the Brauer group of $X$ at local points. If an element of the Brauer group has order coprime to $p$, then its evaluation at a $p$-adic point factors via reduction of the point modulo $p$. For elements of order a power of $p$, this is no longer the case: in order to compute the evaluation map one must know the point to a higher $p$-adic precision. Classifying Brauer group elements according to the precision required to evaluate them at $p$-adic points gives a filtration which we describe using work of Kato. Applications of our work include addressing Swinnerton-Dyer's question about which places can play a role in the Brauer–Manin obstruction. This is joint work with Martin Bright.

number theory

Audience: researchers in the topic

Harvard number theory seminar