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

基本信息

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

项目摘要

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.

该奖项为研究一般以及特殊（主要是全局）域上的代数群有理点群的正态子群结构的研究提供资金。主要研究者试图证明任意无限域上简单代数群的有理点群的有限商的可解性的新猜想。他还计划分析有限维除代数的乘法群的有限商，以获得它们的合理分类。另一个核心问题是对全球域上特殊酉群的马古利斯-普拉托诺夫猜想的研究。首席研究员与 Gopal Prasad、Yoav Segev 和 Gary Seitz 合作解决这些问题。他还计划与 Gopal Prasad 合作编写关于同余子群问题的合着书项目。与线性群的正规子群结构相关的问题有 19 世纪的历史根源（Galois、Jordan、Dixon），并且一直是当今活跃研究的领域。 20世纪。最近，首席研究员与 Y.Segev 和 G.Seitz 的联合工作中发现了分析一般场上各向异性群问题的新技术。这些技术使人们能够使用评估理论的方法来研究一般领域的群体，而这些方法以前仅在全球领域的数论环境中使用。这种方法适合同余子群问题的一般研究领域，该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相关。