Normal Subgroup Structure of the Groups of Rational Points of Algebraic Groups and of Their Special Subgroups

代数群及其特殊子群有理点群的正规子群结构

• 批准号: 0138315
• 负责人: Andrei Rapinchuk
• 金额: $ 12.4万
• 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: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0138315&HistoricalAwards=false
• 关键词: Normal; Subgroup; Structure; Groups; Rational

This award provides funding for an investigation of the normal subgroupstructure of groups of rational points of algebraic groups overgeneral as well as over special (primarily, global) fields. Theprincipal investigator attempts to prove a new conjecture onsolvability of finite quotients of the groups of rational points ofsimple algebraic groups over arbitrary infinite fields.He also plans to analyze finite quotients of the multiplicative groupof a finite dimensional division algebra in order to obtain theirreasonable classification. Another central problem is investigationof the Margulis-Platonov conjecture for special unitary groups overglobal fields. The principal investigator collaborates on these problemswith Gopal Prasad, Yoav Segev and Gary Seitz. He also plans to workwith Gopal Prasad on a joint book project in the congruence subgroupproblem.Questions related to the normal subgroup structure of linear groupshave historical roots in the 19th century (Galois, Jordan, Dixon),and have been an area of active research in the 20th century. Recently,new techniques for analyzing this problem for anisotropic groupsover general fields have been discovered in the joint work of theprincipal investigator with Y.Segev and G.Seitz. These techniques enableone to investigate groups over general fields using methods of thetheory of valuations, which were previously used only in thenumber-theoretic setting of global fields. This approach fits intothe general area of investigation of the congruence subgroup problem,which is connected with other fundamental problems in number theory,currently applied in data transmission, data processing and communicationsystems.

该奖项提供资金的调查正常subgroupstructure群体的合理点的代数群体overgeneral以及在特殊（主要是全球）领域。主要研究者试图证明任意无限域上单代数群的有理点群的有限直积的可解性的一个新猜想，并计划分析有限维除代数的乘法群的有限直积，以得到它们的不合理分类。另一个中心问题是研究整体域上特殊酉群的Margulis-Platonov猜想。主要研究者与Gopal Prasad、Yoav Segev和加里塞茨合作研究这些问题。他还计划与Gopal Prasad合作开展一个关于同余子群问题的联合图书项目。与线性群的正规子群结构相关的问题历史根源于19世纪（伽罗瓦、乔丹、狄克逊），并且一直是一个活跃的研究领域在20世纪。最近，在首席研究员与Y.Segev和G. Seitz的联合工作中，发现了分析一般域上各向异性群的新技术。这些技术使leone研究群体一般领域使用的方法thetheory的估值，这是以前只使用在theber-theoretical设置的全球领域。这种方法适用于研究同余子群问题的一般领域，它与数论中的其他基本问题有关，目前应用于数据传输、数据处理和通信系统。