Shimura Varieties, the Trace Formula, Congruences and Galois Representations

志村簇、迹公式、同余式和伽罗瓦表示法

• 批准号: 0071404
• 负责人: Freydoon Shahidi
• 金额: $ 4.2万
• 依托单位: Purdue University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-05-15 至 2000-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0071404&HistoricalAwards=false
• 关键词: Shimura; Varieties; Trace; Formula; Congruences

Shimura varieties, the trace formula, congruences and Galois representationsStephen S. Kudla (University of Maryland)Freydoon Shahidi (Purdue University)This project will provide support allowing young researchers from the US mathematical community to benefit from participation in the special program at the Institute Henri Poincare (IHP) in Paris in the spring semester 2000. This program focuses on two topics: (i) Shimura varieties and the trace formula and (ii) congruences and Galois representations. These topics, and particularly their interaction, will certainly be at the center of much of the research activity in automorphic forms and number theory in the opening decades of the 21st century. The activity at IHP will bring together the world leaders in these areas. The program will center around a series of lecture `courses' covering the latest developments concering the trace formula, endoscopy, the fundamental lemma, L functions for Shimura varieties, global and local Langlands functoriality, Galois representations, p-adic Hecke algebras, p-adic modular forms, rigid analysis, the local Langlands correspondence and the geometric Langlands correspondence. The scope of the program encourages new directions for research at the interface of the two major fields and participation will provide young researchers a unique opportunity to develop expertise in this important area at an early stage in their careers. Two major developments in mathematics in the later part of the 20th century are the Langlands program in automorphic forms/representation theory and the Wiles and Taylor-Wiles proof of Fermat's Last Theorem and the Taniyama-Shimura conjecture. These advances, relating number theory and geometry, are in fact very closely linked, and a vigorous development of the union of the techniques from the two areas is currently taking place. The resulting field will be one of the main arenas of research activity in mathematics in the first decades of the 21st century. The research program taking place at the Institute Henri Poincare in Paris in the spring semester 2000 and centered around lecture courses by the world leaders provides an unparalleled level of vision and insight. This NSF Grant award will provide funding for young researchers from the US mathematical commmunity to participate in the IHP program, and hence will help to ensure a strong level of US expertise in these new developments in number theory. This award is being supported by the Division of Mathematical Sciences (Algebra and Number Theort program), the Divison of International Programs (Western Europe Program), and the Office of Multidisciplinary Activities of the Mathematical and Physical Sciences Directorate .

志村变量、迹公式、同余式和伽罗瓦表示。这个项目将为美国数学界的年轻研究人员提供支持，使他们能够参加2000年春季学期在巴黎庞加莱研究所（IHP）的特别项目。本课程主要关注两个主题：(i)志村变量和迹公式；（ii）同余和伽罗瓦表示。这些主题，特别是它们之间的相互作用，必将成为21世纪头几十年自同态形式和数论研究活动的中心。国际卫生计划的活动将把这些领域的世界领导人聚集在一起。该计划将围绕一系列讲座“课程”展开，涵盖有关迹公式，内视镜，基本引理，志村变量的L函数，全局和局部朗兰兹泛函，伽罗瓦表示，p进Hecke代数，p进模形式，刚性分析，局部朗兰兹对应和几何朗兰兹对应的最新发展。该计划的范围鼓励在两个主要领域的界面上研究新的方向，参与将为年轻的研究人员提供一个独特的机会，在他们职业生涯的早期阶段发展这一重要领域的专业知识。20世纪后半叶数学的两个主要发展是自同构形式/表示理论中的朗兰兹程序、费马大定理的Wiles和Taylor-Wiles证明以及谷山-志村猜想。这些有关数论和几何的进展，实际上是紧密相连的，目前这两个领域的技术正在蓬勃发展。由此产生的领域将是21世纪头几十年数学研究活动的主要领域之一。该研究项目于2000年春季学期在巴黎的亨利·庞加莱研究所进行，以世界各国领导人的讲座课程为中心，提供了无与伦比的视野和洞察力。美国国家科学基金会将为美国数学界的年轻研究人员提供资金，以参与IHP计划，从而有助于确保美国在数论的这些新发展方面具有强大的专业知识水平。该奖项由数学科学部（代数和数论项目）、国际项目部（西欧项目）以及数学和物理科学理事会的多学科活动办公室提供支持。