Mathematical Sciences: Cohomology of Arithmetic Groups

Ash 9531675该奖项为一个项目提供资金，该项目解决了整数矩阵组与整数系数多项式方程系统的数论方面之间的一些联系。特别是，它研究算术群的上同调作为一个模块的赫克代数，与连接的动机和伽罗瓦表示和自守形式的阿黛尔群。本文研究了一般算术群的上同调类的p-adic族理论，特别是它对GL（m）中有限指数子群的应用。还研究了一系列计算机计算，旨在探讨存在的非自对偶自守表示的上同调型，另一系列测试猜想有关模伽罗瓦表示Hecke本征类在GL（3，Z）与扭曲系数mod p.这个项目属于一般领域的算术代数几何福尔斯-一门融合了两个最古老的数学领域：数论和几何的学科。 事实证明，这种结合非常富有成效--最近解决了几代人都无法解决的问题。在它的许多后果是新的纠错码。 此类代码对于现代计算机（硬盘）和光盘都是必不可少的。

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.