Mathematical Sciences: Cohomology of Arithmetic Groups

这个奖项为一个项目提供资金，该项目解决了积分矩阵组之间的一些联系和整数系数多项式方程系统的数论方面的问题。特别地，它研究了算术群的上同调，作为Hecke代数的一个模块，它与动机和伽罗瓦表示以及在adele群上的自同构形式有联系。研究了一般算术群上同类的p进族理论，并特别应用于GL(m)中有限指数的子群。还研究了一系列计算机计算，旨在探索上同调类型的非自对偶自同构表示的存在性，以及另一系列用于测试与GL（3，Z）具有扭曲系数mod p的上同调中的Hecke特征类的模伽罗瓦表示相关的猜想。该项目属于算术代数几何的一般领域-一门融合了数学中两个最古老领域的学科：数论和几何。事实证明，这一组合非常富有成效——最近解决了几代人经受住了考验的问题。其诸多后果之一是新的纠错码。这些代码对现代计算机（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.