Mathematical Sciences: Cohomology of Arithmetic Groups

数学科学：算术群的上同调

• 批准号: 9531675
• 负责人: Avner Ash
• 金额: --
• 依托单位: Ohio State University Research Foundation -DO NOT USE
• 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: 1996-07-01 至 2000-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9531675&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Cohomology; Arithmetic; Groups

Ash 9531675 This award provides funding for a project that addresses some of the connections between groups of integral matrices and the number theoretical aspects of systems of polynomial equations with integer coefficients. In particular, it studies the cohomology of arithmetic groups as a module for the Hecke algebra, with connections to motives and Galois representations and automorphic forms on adele groups. A theory of p-adic families of cohomology classes for general arithmetic groups, with particular application to subgroups of finite index in GL(m) is considered. Also investigated is a series of computer calculations designed to probe for the existence of non-selfdual automorphic representations of cohomological type, and another series to test a conjecture relating modular Galois representations to Hecke eigenclasses in the cohomology of GL(3,Z) with twisted coefficients mod p. This project falls into the general area of Arithmetic Algebraic Geometry -a subject that blends two of the oldest areas of mathematics: Number Theory and Geometry. This combination has proved extraordinarily fruitful - having recently solved problems that withstood generations. Among its many consequences are new error correcting codes. Such codes are essential for both modern computers (hard disks) and compact disks.

Ash 9531675 该奖项为一个项目提供资金，该项目解决积分矩阵组之间的一些联系以及具有整数系数的多项式方程组的数论方面。特别是，它研究算术群的上同调作为赫克代数的模块，与动机和伽罗瓦表示以及阿黛尔群的自守形式有关。考虑一般算术群的上同调类的 p 进族理论，特别是应用于 GL(m) 中有限指数子群。还研究了一系列计算机计算，旨在探测上同调类型的非自对偶自守表示的存在性，以及另一个系列来测试将模伽罗瓦表示与 GL(3,Z) 上同调中的 Hecke 本征类相关的猜想与扭曲系数 mod p。 该项目属于算术代数几何的一般领域，该学科融合了两个最古老的数学领域：数论和几何。 事实证明，这种结合非常富有成效——最近解决了几代人都面临的问题。其众多后果之一就是新的纠错码。 此类代码对于现代计算机（硬盘）和光盘都是必不可少的。