The genus of a division algebra and the unramified Brauer group

TL;DR

This paper introduces a method to analyze the genus of a division algebra over a field by relating it to the unramified Brauer group, providing tools to prove finiteness and estimate its size.

Contribution

It presents a general approach linking the genus of a division algebra to the unramified Brauer group, enabling finiteness proofs and size estimates.

Findings

Finiteness conditions for the genus of division algebras

Estimation techniques for the size of the genus

Connection between the genus and unramified Brauer groups

Abstract

Let D be a finite-dimensional central division algebra over a field K. We define the genus gen(D) of D to be the collection of classes in the Brauer group of K represented by central division K-algebras D' having the same maximal subfields as D. In this paper, we describe a general approach to proving the finiteness of gen(D) and estimating its size that involves the unramified Brauer group with respect to an appropriate set of discrete valuations of K.