Normal Subgroups of the Groups of Rational Points of Algebraic Groups, Congruence Subgroup Problem, and Related Topics

University Of Virginia Main Campus, Charlottesville VA

Investigators

Abstract

The principle investigator is studying normal subgroups in the groups of rational points of algebraic groups and in their important subgroups (such as S-arithmetic subgroups). Questions of this nature are rooted in the works of the founders of modern algebra such as Galois, Jordan and Dixon, and have been an area of active research in various periods of the 20th century (among important contributors one can mention Artin, Dieudonne, Tits). While these works dealt mainly with the isotropic case where one can use unipotent elements, the PI's research focuses on the anisotropic case where no unipotent elements are available, hence essentially new techniques are needed. Anisotropic groups are usually associated with noncommutative division algebras. Recently, in a joint work of the PI with Y.Segev and G.M.Seitz, new methods for analyzing normal subgroups of the multiplicative group of a finite dimensional division algebra were developed, and the current proposal describes a variety of problems where these methods or their suitable adaptations can (and will) be used. In particular, the PI intends to make a substantial progress in the investigation of unitary groups over global as well as general fields. Another central topic of the project is the congruence subgroup problem for S-arithmetic groups. The PI will continue the ongoing joint research with G.Prasad focused on proving centrality of the congruence kernel in new cases, and also the work on the book project devoted to the congruence subgroup problem. The investigator's research is on the structure of classical and algebraic groups and their arithmetic subgroups. Questions of this nature are rooted in the works of the founders of modern algebra such as Galois, Jordan and Dixon, and have been an area of active research in various periods of the 20th century. In particular, it should be noted that the congruence subgroup problem is connected with other fundamental problems in number theory, currently applied in data transmission, data processing and communication systems.