Foundations of the Relative Langlands Program

朗兰兹程序是现代数论的核心，其引人注目的猜想主导了该领域的大部分内容。它的主要预测包括丢番图方程世界和称为自同构形式的分析对象世界之间的联系(称为互易性)，以及后者之间令人惊讶的关系(称为函数性)。目前的项目集中在这些预测中的第二个，尽管近几十年来已经进行了大量的研究，但缺乏适当的理解(功能化的确切性质是什么？我们应该比较的对象是什么？)或其证据的策略(除了用Arthur-Selberg轨迹公式证明的极其重要但相对有限的“内窥镜检查”实例之外)。这个项目将为推广这些猜想奠定基础，称为“相对功能性”如下：要比较的对象是代数几何空间上的施瓦茨函数，称为“堆栈”，比较它们的方法将沿着朗兰兹提出的“超越内窥镜”思想的思路来研究。更准确地说，要考虑的代数堆栈是由一对具有约化群G的作用的球面变元产生的。每个变元上都有一个L函数，推广了兰金-塞尔伯格和周期积分的方法；这个L的作用将在PI和其他人早期工作的基础上进行研究。Jacquet的相对迹公式将被进一步发展，将Arthur-Selberg迹公式推广为G的对角作用在这些变元的商上的分布。从低阶情形出发，研究人员将通过积分变换来检验比较相对迹公式的新的“非标准”方法。在这一过程中，齐次空间上的调和分析将得到新的结果。

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.