Mathematical Sciences: Group Theory Methods in Number Theory

数学科学：数论中的群论方法

• 批准号: 9622590
• 负责人: Nigel Boston
• 金额: $ 7.8万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-05-15 至 1999-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9622590&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Group; Theory; Methods

9622590 Boston This award is for a project connecting Group Theory and Number Theory. Many links between these two areas have already been forged, in particular, the use of Galois representations, whose study, by group-theoretic means or otherwise, has consequences to the theory of elliptic curves and consequently to Fermat's Last Theorem. In a prior supported project, the PI exploited this link to prove theorems on the deformations of Galois representations and the structure of Galois groups, particularly of unramified extensions. This project will exploit these ideas further with an eye towards proving further cases of the Fontaine-Mazur conjecture and towards elucidating the structure of deformation spaces of Galois representations. The investigator also plans to continue his exploration of the function field case. Finally, the investigator will study Dirichlet series attached to profinite groups. This part of the project provides plenty of opportunity to transfer information on groups (much obtained computationally) to information in number theory, and vice versa. This research falls into the general mathematical fields of Number Theory and Group Theory. Group Theory is the study of groups, which are algebraic structures with a single operation. It appears in many areas of mathematics, as well as physics and chemistry. Number theory has its historical roots in the study of the whole numbers, addressing such questions as those dealing with the divisibility of one whole number by another. It is among the oldest branches of mathematics and was pursued for many centuries for purely aesthetic reasons. However, within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission, data processing, and communication systems.

9622590 Boston这个奖项是颁给一个连接群论和数论的项目。这两个领域之间的许多联系已经形成，特别是伽罗瓦表示的使用，其研究，通过群论手段或其他方式，对椭圆曲线理论产生了影响，从而对费马大定理产生了影响。在先前支持的一个项目中，PI利用这一联系证明了伽罗瓦表示的变形和伽罗瓦群的结构，特别是无分支扩展的定理。该项目将进一步利用这些想法，以证明Fontaine-Mazur猜想的进一步案例，并阐明伽罗瓦表示的变形空间结构。研究者还计划继续他对函数场案例的探索。最后，研究者将研究无限群上的狄利克雷级数。项目的这一部分提供了大量的机会，可以将关于群体的信息（通过计算获得的）转换为数论中的信息，反之亦然。本研究属于数论和群论的一般数学领域。群论是研究群的学科，群是具有单一运算的代数结构。它出现在数学的许多领域，以及物理和化学。数论的历史根源在于对整数的研究，解决的问题是一个整数能被另一个整数整除的问题。它是数学中最古老的分支之一，人们为了纯粹的美学原因而追求了许多世纪。然而，在过去的半个世纪里，它已经成为数据传输、数据处理和通信系统等各种应用领域不可或缺的工具。