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具有整数系数的多项式方程系统的数量理论方面之间的一些联系。特别地，作为Hecke代数的模来研究算术群的上同调，它与Adele群上的动机、Galois表示和自同构形式有联系。考虑了一般算术群上同调类的p-进数族理论，特别地应用于GL(M)中有限指标子群。还研究了一系列计算机计算，旨在探索上同调型的非自对偶自同构表示的存在，以及另一系列测试一个猜想，该猜想将模伽罗瓦表示与具有扭曲系数mod p的GL(3，Z)的上同调中的Hecke特征类联系起来。本课题属于算术代数几何的一般领域--这是一门融合了数论和几何这两个最古老的数学领域的学科。事实证明，这种结合非常有成效--最近解决了几代人经受住的问题。在其众多后果中，有一种是新的纠错码。这种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.