Eisenstein Series, Continuous Spectrum, and the Relative Trace Formula

艾森斯坦级数、连续谱和相对痕量公式

• 批准号: 9700950
• 负责人: Jonathan Rogawski
• 金额: $ 7.5万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1997
• 资助国家: 美国
• 起止时间: 1997-08-01 至 2000-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9700950&HistoricalAwards=false
• 关键词: Eisenstein; Series; Continuous; Spectrum; Relative

Rogawski 9700950 This award funds the research of Professor Rogawski, who is interested in developing the relative trace formula for a pair (G,H) consisting of a quasi-split reductive group G and subgroup H, the fixed-point set of an involution. The PI intends to investigate problems connected with the contribution of the continuous spectrum to the relative trace formula. The first goal of this project is the study of period integrals of truncated Eisenstein series over H . The PI intends to develop a theory of such integrals, and in particular, explicit formulas extending his current work with Jacquet on GL(n). The second goal is to evaluate the contribution of the spectral kernel to the relative trace formula for G. These integrals will be evaluated using truncation, the explicit formulas, and a suitable theory of (G,M) -families. These results will be applied as one ingredient in the comparison of the relative trace formula with the Kuznetzov trace formula in certain cases. This proposal is in the part of mathematics known as the Langlands program. This program represents a fusion of Number Theory and Representation Theory , and it has been a stimulus to a great deal of recent research in both fields. Number Theory is one of the oldest branches of mathematics and is concerned with the most basic of mathematical objects, the ordinary whole numbers. However, it turns out that in order to express many of the patterns and relations discovered by mathematicians, it is necessary to use some of the most advanced and technical theories of twentieth century mathematics. On the other hand, the problems of number theory have provided a powerful stimulus to research in other diverse parts of the discipline. The Langland's program provides a framework for investigating and vastly generalizing the so-called reciprocity laws of number theory using the tools of infinite-dimensional representation theory. Although very technical and deep, this program has found astonishing applica tions in areas like theoretical computer science (construction of expanding graphs) and coding theory (finding optimal Goppa codes). It has also played an indispensable role in some of the recent spectacular developments in number theory itself, such as the proof of Fermat's Last Theorem.

Rogawski 9700950 该奖项资助了Rogawski教授的研究，他有兴趣开发由准分裂还原群G和子群H组成的对（G，H）的相对迹公式，子群H是对合的不动点集。 PI旨在研究与连续谱对相对迹公式的贡献有关的问题。 本项目的第一个目标是研究环H上截断Eisenstein级数的周期积分。PI打算发展这样的积分理论，特别是显式公式，扩展他目前与Jacquet在GL（n）上的工作。 第二个目标是评估谱核对G的相对迹公式的贡献。 这些积分将使用截断、显式公式和适当的（G，M）族理论来计算。 在某些情况下，这些结果将作为比较相对迹公式与库兹涅佐夫迹公式的一个组成部分。 这个建议是在数学的一部分被称为朗兰兹纲领。 这个程序代表了数论和表示论的融合，它已经刺激了这两个领域的大量研究。数论是数学中最古老的分支之一，涉及最基本的数学对象，即普通的整数。 然而，事实证明，为了表达数学家发现的许多模式和关系，必须使用二十世纪数学的一些最先进和技术性的理论。 另一方面，数论的问题为该学科其他不同部分的研究提供了强有力的刺激。 朗格兰的程序提供了一个框架，用于使用无限维表示论的工具来研究和广泛推广数论的所谓互易定律。 虽然非常技术性和深度，这个程序已经在理论计算机科学（构建扩展图）和编码理论（寻找最佳Goppa码）等领域找到了惊人的应用。 它也在数论本身最近的一些引人注目的发展中发挥了不可或缺的作用，例如费马大定理的证明。