On the Discrete Spectrum of Classical Groups and Converse Theorems

这个研究项目涉及朗兰兹纲领中的一些问题，朗兰兹纲领是罗伯特·朗兰兹在20世纪60年代提出的一个纲领。朗兰兹程序是一个影响深远的猜想网，它预测了算术（例如，多项式方程解的性质）和分析（例如，对称流形上某些微分方程的高度对称解，称为自同构形式）之间的惊人联系。例如，著名的怀尔斯对费马大定理的证明，使用了朗兰兹和Tunnell证明的朗兰兹程序的早期结果。在另一个方向上，自同构形式与弦理论和物理学中的黑洞研究有着深刻的联系。在这个项目中，PI将研究自同构形式的解析性质及其在朗兰兹程序中的数论结果。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，随后许多人将经典群的离散谱划分为由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包中模块的具体构造；每个Arthur包中自同构表示的傅里叶系数，包括Jiang猜想每个Arthur包的私密性；不同但密切相关的群体之间的Arthur包关系（通过自同构血统）。项目的第二部分是关于逆向问题。逆问题旨在从它们的傅里叶系数中恢复模/自同构形式。例如，著名的Hecke和Weil的逆定理给出了Dirichlet级数是模形式的Mellin变换的充分条件。已知逆定理在朗兰泛函的建立中起着重要的作用。在项目的这一部分，PI将开发几种猜想的方法，包括Jacquet猜想和Cogdell-Piatetski-Shapiro猜想，以证明最优局部和全局逆定理。

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.