Mathematical Sciences: Arithmetic Models for Shimura Varieties, L-Functions and Cohomology Groups as Integral Representations

9623269 Pappas The PI将继续他在以下两个问题上的工作：a.利用齐次空间的等变嵌入理论的思想，研究非光滑归约素数上Shimura簇的算术模型的几何和局部性质。B.研究具有有限群作用的算术簇的上同调中出现的积分表示。这是将A.Fr；3hlich和M.Taylor的整数环的Galois模结构理论推广到高维算术簇的程序的一部分。作者建议研究描述这些表示的不变量与从L函数获得的数据的关系。这是在代数几何领域的研究。代数几何是现代数学中最古老的部分之一，但在过去的25年里，它已经取得了革命性的成就。在它的起源中，它处理的是可以在平面上用最简单的方程定义的图形，即多项式。如今，该领域不仅使用代数的方法，而且使用分析和拓扑学的方法，反过来，在这些领域以及物理、理论计算机科学和机器人中也找到了应用。

9623269 Pappas The PI will continue his work on the following two problems: A. Study the geometry and the local properties of arithmetic models of Shimura varieties over primes of non-smooth reduction, by using ideas from the theory of equivariant embeddings of homogeneous spaces. B. Study the integral representations that appear in the cohomology of arithmetic varieties with finite group actions. This is part of a program for extending the theory of Galois module structure of rings of integers of A. Fr;3hlich and M. Taylor to higher dimensional arithmetic varieties. The author proposes to investigate relations of invariants describing these representations with data obtained from L-functions. This is research in the field of algebraic geometry. Algebraic geometry is one of the oldest parts of modern mathematics, but one which has had a revolutionary flowering in the past quarter-century. In its origin, it treated figures that could be defined in the plane by the simplest equations, namely polynomials. Nowadays the field makes use of methods not only from algebra, but from analysis and topology, and conversely is finding application in those fields as well as in physics, theoretical computer science, and robotics.