On the Discrete Spectrum of Classical Groups and Converse Theorems

这项研究项目涉及朗兰兹计划中的某些问题，朗兰兹计划是罗伯特·朗兰兹在20世纪60年代提出的计划。朗兰兹计划是一个影响深远的猜想网络，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，称为自同构形式)之间惊人的联系。例如，A.Wiles的著名的费马大定理的证明，使用了朗兰兹和图内尔证明的朗兰兹计划中的早期结果。另一方面，自同构形式与弦理论和物理学中对黑洞的研究有很深的联系。在这个项目中，PI将在朗兰兹计划中研究自同构形式的解析性质及其数论结果。自同构形式理论的一个主要主题是研究定义在数域上的连通约化群的离散谱。通过Arthur的开创性工作以及其他许多人的工作，经典群的离散谱被分类为由Arthur参数参数化的Arthur包。本课题的第一部分是对Arthur包的精细结构进行分析，包括：每个Arthur包中模的具体构造；每个Arthur包中自同构表示的傅立叶系数，包括江氏猜想；每个Arthur包的尖端性；以及不同但密切相关的群之间的关系(通过自同构下降)。该项目的第二部分是关于逆问题的。逆问题的目标是从傅立叶系数恢复模/自同构形式。例如，著名的Hecke和Weil逆定理给出了Dirichlet级数是模形式的Mellin变换的充分条件。众所周知，逆定理在建立朗兰兹函数中起着重要的作用。在项目的这一部分，PI将开发几个猜想的方法，包括Jacquet猜想和Cogdell-Piatetski-Shapiro猜想，以证明最优的局部和整体逆定理。

This research project concerns certain problems within the Langlands Program, a program proposed by Robert Langlands in 1960s. The Langlands Program is a web of far-reaching and influential conjectures that 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, known as automorphic forms). The celebrated proof of Fermat's Last Theorem by A. Wiles, for instance, uses early results in the Langlands program proved by Langlands and Tunnell. In another direction, automorphic forms have deep connections with the string theory and the study of black holes in physics. In this project the PI will investigate analytic properties of automorphic forms and their number-theoretic consequences in the Langlands program. A main theme in the theory of automorphic forms is to study the discrete spectrum of connected reductive groups defined over number fields. By the pioneered work of Arthur, followed by many others, the discrete spectrum of classical groups has been classified into Arthur packets parametrized by Arthur parameters. The first part of this project is to analyze the finer structure of Arthur packets, including: concrete constructions of modules in each Arthur packet; Fourier coefficients of automorphic representations in each Arthur packet, including Jiang's conjecture; cuspidality of each Arthur packet; and relations among Arthur packets of different but closely related groups (via automorphic descent). The second part of the project is about converse problems. Converse problems aim to recover modular/automorphic forms from their Fourier coefficients. For example, the famous converse theorems of Hecke and Weil give sufficient conditions for a Dirichlet series to be the Mellin transform of a modular form. It is known that converse theorems play an important role in the establishment of Langlands functoriality. In this part of the project the PI will develop approaches to several conjectures, including Jacquet's conjecture and Cogdell-Piatetski-Shapiro conjecture, in order to prove optimal local and global converse theorems.