Igor A. Rapinchuk

Division algebras with the same maximal subfields

We give a survey of recent results related to the problem of characterizing finite-dimensional division algebras by the set of isomorphism classes of their maximal subfields. We also discuss various generalizations of this problem and some of its applications. In the last section, we extend the problem to the context of absolutely almost simple algebraic groups.

On linear representations of Chevalley groups over commutative rings

Let

The genus of a division algebra and the unramified Brauer group

G

Centrality of the congruence kernel for elementary subgroups of Chevalley groups of rank

be the universal Chevalley-Demazure group scheme corresponding to a reduced irreducible root system of rank

> 1

\geq 2

over noetherian rings

, and let

On the size of the genus of a division algebra

R

ON ABSTRACT HOMOMORPHISMS OF CHEVALLEY GROUPS OVER THE COORDINATE RINGS OF AFFINE CURVES

be a commutative ring. We analyze the linear representations

On abstract representations of the groups of rational points of algebraic groups in positive characteristic

\rho \colon G(R)^+ \to GL_n (K)

On the genus of a division algebra

over an algebraically closed field