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

主要研究者正在研究代数群的有理点群及其重要子群（例如S-算术子群）中的正规子群。这种性质的问题植根于现代代数创始人如伽罗瓦、乔丹和狄克逊的著作中，并且一直是 20 世纪各个时期的一个活跃的研究领域（重要贡献者之一可以提到 Artin、Dieudonne、Tits）。虽然这些工作主要涉及可以使用单能元件的各向同性情况，但 PI 的研究重点是没有单能元件可用的各向异性情况，因此本质上需要新技术。各向异性群通常与非交换除代数相关。最近，在 PI 与 Y.Segev 和 G.M.Seitz 的联合工作中，开发了用于分析有限维除代数乘法群的正规子群的新方法，当前的提案描述了可以（并将）使用这些方法或其适当改编的各种问题。特别是，PI打算在全球和一般领域的酉群研究方面取得实质性进展。该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.Prasad 进行持续的联合研究，重点是证明同余核在新案例中的中心地位，以及致力于同余子群问题的图书项目的工作。研究者的研究是经典群和代数群及其算术子群的结构。这种性质的问题植根于现代代数创始人伽罗瓦、乔丹和狄克逊的著作中，并且一直是 20 世纪各个时期的活跃研究领域。特别值得注意的是，同余子群问题与数论中的其他基本问题相关，目前应用于数据传输、数据处理和通信系统。

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.