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

代数群有理点群的正规子群、同余子群问题及相关主题

• 批准号: 0502120
• 负责人: Andrei Rapinchuk
• 金额: $ 20.67万
• 依托单位: University of Virginia Main Campus
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-06-01 至 2011-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0502120&HistoricalAwards=false
• 关键词: Normal; Subgroups; Groups; Rational; Points

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.

主要研究者是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。这种性质的问题是植根于工程的创始人现代代数，如伽罗瓦，约旦和狄克逊，并一直是一个领域的积极研究在各个时期的20世纪世纪（其中重要的贡献者之一，可以提到阿廷，迪厄多内，山雀）。虽然这些工作主要是处理各向同性的情况下，可以使用单幂元素，PI的研究重点是各向异性的情况下，没有单幂元素，因此基本上需要新的技术。各向异性群通常与非交换除代数相联系。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的各种问题。特别是，PI打算在全球和一般领域的调查中取得实质性进展。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正在进行的联合研究，重点是证明新情况下的同余核的中心性，以及致力于同余子群问题的书籍项目的工作。调查员的研究是对结构的经典和代数群体和他们的算术分组。这种性质的问题是植根于工程的创始人现代代数，如伽罗瓦，约旦和狄克逊，并一直是一个领域的积极研究在各个时期的20世纪。特别是，应该注意的是，同余子群问题与目前应用于数据传输，数据处理和通信系统的数论中的其他基本问题有关。