The genus of a division algebra and the unramified Brauer group

Abstract

Let \(D\) be a finite-dimensional central division algebra over a field \(K\). We define the genus \(\mathbf{gen}(D)\) of \(D\) to be the collection of classes \([D^{\prime }] \in \mathrm{Br}(K)\), where \(D^{\prime }\) is a central division \(K\)-algebra having the same maximal subfields as \(D\). In this paper, we describe a general approach to proving the finiteness of \(\mathbf{gen}(D)\) and estimating its size that involves the unramified Brauer group with respect to an appropriate set of discrete valuations of \(K\). This approach is then implemented in some concrete situations, yielding in particular an extension of the Stability Theorem of A. Rapinchuk and I. Rapinchuk (Manuscr. Math. 132:273–293, 2010) from quaternion algebras to arbitrary algebras of exponent two. We also consider an example where the size of the genus can be estimated explicitly. Finally, we offer two generalizations of the genus problem for division algebras: one deals with absolutely almost simple algebraic \(K\)-groups having the same isomorphism/isogeny classes of maximal \(K\)-tori, and the other with the analysis of weakly commensurable Zariski-dense subgroups.