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

主要研究代数群的有理点群及其重要子群（如s -算术子群）中的正规子群。这种性质的问题根植于现代代数奠基人的著作中，如伽罗瓦、乔丹和迪克森，并且在20世纪的各个时期一直是一个活跃的研究领域（在重要的贡献者中，人们可以提到Artin、Dieudonne和Tits）。虽然这些工作主要是处理各向同性的情况下，可以使用单幂元，但PI的研究重点是各向异性的情况下，没有可用的单幂元，因此本质上需要新的技术。各向异性群通常与非交换除法代数有关。最近，在PI与Y.Segev和G.M.Seitz的联合工作中，开发了分析有限维除法代数的乘法群的正规子群的新方法，并且当前的提议描述了可以（并且将）使用这些方法或其适当的适应性的各种问题。特别是，调查团计划不仅对一般领域进行调查，而且还对全球范围内的单一制团体进行调查，并取得实质性进展。项目的另一个中心主题是s -算术群的同余子群问题。PI将继续与g.p拉萨德进行的联合研究，重点是在新情况下证明同余核的中心性，以及致力于同余子群问题的书籍项目的工作。研究者的研究方向是经典群和代数群及其算术子群的结构。这种性质的问题根植于现代代数奠基人的著作中，如伽罗瓦、乔丹和迪克森，并且在20世纪的各个时期一直是一个活跃的研究领域。需要特别指出的是，同余子群问题与数论中的其他基本问题相联系，目前应用于数据传输、数据处理和通信系统中。

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 the20th 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 withnoncommutative 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 ofunitary groups over global as well as general fields. Anothercentral topic of the project is the congruence subgroup problemfor S-arithmetic groups. The PI will continue the ongoing jointresearch with G.Prasad focused on proving centrality of thecongruence kernel in new cases, and also the work on the bookproject 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.