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

数学科学：Shimura 簇、L 函数和上同调群的算术模型作为积分表示

• 批准号: 9623269
• 负责人: Georgios Pappas
• 金额: $ 7.95万
• 依托单位: Princeton University
• 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 1999-08-24
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9623269&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Arithmetic; Models; Shimura

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.

9623269 Pappas PI 将继续解决以下两个问题： A. 利用齐次空间等变嵌入理论的思想，研究非平滑约简素数上 Shimura 簇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 整数环的伽罗瓦模结构理论扩展到更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。 这是代数几何领域的研究。代数几何是现代数学最古老的部分之一，但在过去的四分之一个世纪里得到了革命性的发展。 在它的起源中，它处理可以通过最简单的方程（即多项式）在平面上定义的图形。如今，该领域不仅使用代数方法，还使用分析和拓扑方法，相反，它正在这些领域以及物理学、理论计算机科学和机器人技术中得到应用。