The genus of a division algebra and the unramified Brauer group

2013/03/02 by Chernousov, Vladimir I., Rapinchuk, Andrei S., Rapinchuk, Igor A.
#Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT) #Rings and Algebras (math.RA)

paper · doi:10.48550/arxiv.1303.0390

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.

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