Trace Formula, Analytic Number Theory, and Langlands Functoriality

朗兰兹纲领可以追溯到R. P.朗兰兹到A. Weil在1967年预测了算术（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的高度对称解（即，自守形式）。“功能性猜想”是朗兰兹纲领的核心。这些都是深刻的programmures具有深远的影响，无论是在数论和理论的自守形式。例如，著名的费马大定理的证明。怀尔斯使用了一个由朗兰兹和通内尔证明的函子性假设的例子。虽然在这些问题上取得了重大进展，但就其最大的一般性而言，这些问题仍然是开放的。本研究项目旨在开发工具和技术，以证明更多的情况下，功能性假设。在自守形式理论中最普遍和最有力的工具之一是阿瑟-塞尔伯格迹公式。它已成功地用于证明函数性定理。在所有这些情况下，都是使用两种不同的迹线公式进行比较。在最近的一项名为“超越内窥镜”的提案中，朗兰兹提出了一种新的方法来攻击一般的功能性假设。这是一种全新的方法，旨在使用迹公式分析自守L-函数的极点，特别是，它是非比较的。在Beyond Endoscopy中使用迹线公式的离散部分之前，需要解决迹线公式固有的各种困难。这个项目的目的是双重的：首先是解决这些困难的情况下GL（N）（以下PI的早期工作和建议的亚瑟），并得到一个明确的轨迹公式的尖部的频谱。第二个目标是使用得到的公式对各种自守L函数执行Beyond Endoscopy。

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.