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.

这个奖资助了Rogawski教授的研究，他感兴趣的是发展由准分裂约化群G和子群H组成的对（G，H）的相对迹公式，即对合的不动点集。PI打算研究与连续光谱对相对示踪公式的贡献有关的问题。本课题的第一个目标是研究H上截断的爱森斯坦级数的周期积分。PI打算发展这样的积分理论，特别是，扩展他目前与Jacquet在GL(n)上的工作的显式公式。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（G，M）族的适当理论来评估。在某些情况下，这些结果将作为比较相对量公式与库兹涅佐夫量公式的一个因素加以应用。这个建议属于数学中被称为朗兰兹纲领的部分。这个程序代表了数论和表示理论的融合，并且它已经刺激了这两个领域最近的大量研究。数论是数学最古老的分支之一，它关注的是最基本的数学对象——普通整数。然而，事实证明，为了表达数学家发现的许多模式和关系，有必要使用20世纪数学中一些最先进和最具技术性的理论。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大的刺激。Langland的程序提供了一个框架，用于使用无限维表示理论的工具来研究和广泛推广所谓的数论的互易律。虽然非常技术性和深度，这个程序已经发现了惊人的应用领域，如理论计算机科学（构建展开图）和编码理论（寻找最佳的Goppa代码）。它在最近数论本身的一些惊人发展中也发挥了不可或缺的作用，比如费马大定理的证明。