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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.

朗兰兹程序是现代数论的核心，其引人注目的猜想在该领域占据了主导地位。它的主要预测包括丢番图方程世界和自同构形式的分析对象世界之间的联系（称为互易性），以及后者内部的惊人关系（称为泛函性）。目前的项目集中在第二种预测上，尽管近几十年来对它的研究很多，但它缺乏正确的理解(功能的本质究竟是什么？我们应该比较的对象是什么？)或证明它的策略（除了极其重要但相对有限的“内窥镜检查”用Arthur-Selberg的轨迹公式证明的例子）。该项目将为这些猜想的概括奠定基础——被称为“相对功能”——如下：要比较的对象是称为“堆栈”的代数几何空间上的施瓦茨函数，比较它们的方法将沿着朗兰兹提出的“超越内窥镜”思想的路线进行研究。更准确地说，要考虑的代数堆栈是由一对具有约化群g作用的球变体产生的，其中每个变体都有一个l函数，推广了Rankin-Selberg方法和周期积分方法；这个l函数将在PI和其他人早期工作的基础上进行研究。Jacquet的相对示踪公式将进一步发展，推广Arthur-Selberg示踪公式，作为这些变量通过g的对角线作用在商上的分布。从低秩情况开始，研究者将研究通过积分变换比较相对示踪公式的新的“非标准”方法。在此过程中，齐次空间谐波分析将得到新的结果。