Trace Formula, Analytic Number Theory, and Langlands Functoriality

朗兰兹计划，可以追溯到R.P.朗兰兹1967年写给A.Weil的信，它预测了算术(例如，多项式方程的解的性质)和分析(例如，对称流形上的某些微分方程解的高度对称解(即，自同构形式))之间惊人的联系。“函数性猜想”是朗兰兹计划的核心。这些都是深刻的猜想，在数论和自同构形式理论中都有深远的后果。例如，A.Wiles著名的费马大定理证明使用了早些时候由朗兰兹和图内尔证明的函数性猜想的情况。尽管在这些猜想方面已经取得了重大进展，但总的来说，这些猜想是完全开放的。这个研究项目的目的是开发工具和技术来证明函数性猜想的进一步情况。自同构形理论中最通用和最强大的工具之一是Arthur-Selberg迹公式。它已成功地用于证明函数性猜想的情形。在所有这些情况下，使用的是两个不同的示踪公式之间的比较。在最近一项名为“超越内窥镜”的提案中，朗兰兹提出了一种新的方法，从总体上攻击功能性猜想。利用迹公式分析自同构L函数的极点是一种全新的方法，特别是它具有非比较性。在超越内窥镜检查中使用跟踪公式的离散部分之前，需要解决各种固有的困难。这个项目的目的有两个：首先是解决GL(N)的这些困难(遵循Pi的早期工作和Arthur的建议)，并得到关于谱的尖端部分的显式迹公式。第二个目标是使用得到的公式为各种自同构L函数执行Beyond内窥镜。

The Langlands Program, dating back to the letter written by R. P. Langlands to A. Weil in 1967, predicts surprising connections between arithmetic (e.g. properties of solutions to polynomial equations) and analysis (e.g., highly symmetric solutions to certain differential equations on symmetric manifolds (i.e.,automorphic forms)). The "Functoriality Conjectures" lie at the heart of the Langlands program. These are deep conjectures having far-reaching consequences both in number theory and the theory of automorphic forms. For instance, the celebrated proof of Fermat's Last Theorem by A. Wiles uses a case of functoriality conjectures proved earlier by Langlands and Tunnell. Although significant progress has been made towards these conjectures, in their utmost generality they are wide open. This research project aims to develop tools and techniques to prove further cases of functoriality conjectures. One of the most general and power tools in the theory of automorphic forms is the Arthur-Selberg trace formula. It has been successfully used in proving cases of functoriality conjectures. In all of these cases it is a comparison between two different trace formulas that was utilized. In a recent proposal called "Beyond Endoscopy" Langlands proposed a new approach to attack the functoriality conjectures in general. It is a fundamentally new approach aiming to analyze poles of automorphic L-functions using the trace formula and, in particular, is non-comparative. There are various difficulties, intrinsic to the discrete part of the trace formula, that need to be addressed before utilizing it in Beyond Endoscopy. This aim of this project is twofold: First is to address these difficulties in the case of GL(N) (following PI's earlier works and suggestions of Arthur) and get an explicit trace formula on the cuspidal part of the spectrum. The second goal is to use the resulting formula to execute Beyond Endoscopy for various automorphic L-functions.