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Foundations of the Relative Langlands Program

基本信息

批准号：
1502270
负责人：
Ioannis Sakellaridis
金额：
$ 17.25万
依托单位：
Rutgers University Newark
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2015
资助国家：
美国
起止时间：
2015-07-01 至 2018-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1502270&HistoricalAwards=false
关键词：
Foundations Relative Langlands Program

项目摘要

The Langlands program is at the heart of modern number theory, and its striking conjectures have dominated much of the field. Its main predictions include a connection (called reciprocity) between the world of diophantine equations and a world of analytic objects called automorphic forms, and surprising relations (called functoriality) within the latter. The present project focuses on the second of these predictions which, as much as it has been studied in recent decades, is lacking a proper understanding (what exactly is the nature of functoriality? what are the objects that we are supposed to compare?) or strategy for its proof (besides the extremely important, but relatively limited instances of "endoscopy" proved with the Arthur-Selberg trace formula). The project will establish the foundations for a generalization of these conjectures -- termed "relative functoriality" -- as follows: the objects to be compared are Schwartz functions on algebro-geometric spaces called "stacks," and ways to compare them will be investigated along the lines of the "beyond endoscopy" ideas set forth by Langlands.More precisely, the algebraic stacks to be considered arise from a pair of spherical varieties with an action of a reductive group G. There is an L-function attached to each of these varieties, generalizing the method of Rankin-Selberg and period integrals; this L-function will be studied building upon earlier work of the PI and others. The relative trace formula of Jacquet will be further developed, generalizing the Arthur-Selberg trace formula, as a distribution on the quotient of these varieties by the diagonal action of G. Starting from low-rank cases, the investigator will examine new, "non-standard" ways of comparing relative trace formulas via integral transforms. Along the way, new results in harmonic analysis on homogeneous spaces will be developed.

朗兰兹纲领是现代数论的核心，其引人注目的成就在该领域占据了主导地位。它的主要预言包括丢番图方程世界和分析对象世界（称为自守形式）之间的联系（称为互易性），以及后者内部令人惊讶的关系（称为函性）。目前的项目集中在这些预测的第二个，因为它已经研究了近几十年来，是缺乏适当的理解（究竟是什么性质的功能性？我们应该比较的对象是什么？）或策略的证明（除了极其重要的，但相对有限的情况下，“内窥镜”证明与阿瑟-塞尔伯格迹公式）。该项目将为推广这些被称为“相对功能性”的结构奠定基础，具体如下：要比较的对象是代数几何空间上称为“栈”的Schwartz函数，并且将沿着Langlands提出的“超越内窥镜”思想的路线来研究比较它们的方法。更确切地说，要考虑的代数栈产生于一对具有约化群G的作用的球面簇。每个变量都有一个L-函数，推广了Rankin-Selberg方法和周期积分;这个L-函数将在PI和其他人的早期工作的基础上进行研究。Jacquet的相对迹公式将得到进一步的发展，推广了Arthur-Selberg迹公式，作为G的对角作用在这些簇的商上的分布。从低秩的情况下，调查员将检查新的，“非标准”的方式比较相对轨迹公式通过积分变换。沿着这条道路，齐性空间上调和分析的新结果将得到发展。