On the Discrete Spectrum of Classical Groups and Converse Theorems

本研究项目涉及朗兰兹纲领中的某些问题，朗兰兹纲领是由罗伯特·朗兰兹于20世纪60年代提出的。朗兰兹纲领是一个具有深远影响力的理论网络，它预测了算术之间令人惊讶的联系（例如，多项式方程的解的性质）和分析（例如，对称流形上某些微分方程的高对称解，称为自守形式）。费马大定理的著名证明。例如，怀尔斯使用了朗兰兹和通内尔证明的朗兰兹纲领的早期结果。在另一个方向上，自守形式与物理学中的弦理论和黑洞研究有着深刻的联系。在这个项目中，PI将研究自守形式的分析性质及其在朗兰兹程序中的数论后果。自守形式理论的一个主要主题是研究数域上连通约化群的离散谱。由亚瑟的开创性工作，随后许多其他人，经典群的离散谱已被归类为亚瑟包参数化的亚瑟参数。这个项目的第一部分是分析亚瑟包的更精细的结构，包括：每个亚瑟包中模块的具体构造;每个亚瑟包中自守表示的傅立叶系数，包括江猜想;每个亚瑟包的尖点性;以及不同但密切相关的群的亚瑟包之间的关系（通过自守下降）。第二部分是关于匡威问题的研究。匡威问题的目标是从它们的傅立叶系数恢复模/自守形式。例如，著名的Hecke和Weil的匡威定理给出了Dirichlet级数是模形式的Mellin变换的充分条件。匡威定理在朗兰兹泛函的建立中起着重要的作用。在项目的这一部分中，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全局匡威定理。

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.