Shimura Varieties, Galois Modules and L-functions

Shimura 簇、伽罗瓦模块和 L 函数

• 批准号: 9970378
• 负责人: Georgios Pappas
• 金额: $ 7.44万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-06-01 至 2003-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970378&HistoricalAwards=false
• 关键词: Shimura; Varieties; Galois; Modules; functions

This project has two parts: The first part concerns the construction of integral models of Shimura varieties over primes of non-smooth reduction. The second part focuses on the study of the integral representations that appear in the cohomology of arithmetic varieties with an action by a finite group; there are connections with special values of L-functions, with the theory of class groups of cyclotomic fields and applications to determining the geometric Galois module structure of lattices of modular forms. This research falls into a modern area of number theory known as arithmetic geometry in which geometric methods are used to solve problems in number theory. Number theory has its historical roots in the study of the natural numbers. It is among the oldest branches of mathematics. Within the last half century it has become an indispensable tool in diverse applications in areas such as data transmission and processing, and communication systems.

该项目分为两个部分：第一部分涉及在非平滑约简素数上构建 Shimura 簇积分模型。第二部分重点研究有限群作用的算术簇上同调中出现的积分表示；与 L 函数的特殊值、分圆域的类群理论以及确定模形式格的几何伽罗瓦模结构的应用有联系。 这项研究属于现代数论领域，称为算术几何，其中几何方法用于解决数论中的问题。 数论的历史根源在于自然数的研究。它是数学最古老的分支之一。在过去的半个世纪中，它已成为数据传输和处理以及通信系统等领域的各种应用中不可或缺的工具。