On the Discrete Spectrum of Classical Groups and Converse Theorems

Purdue University, West Lafayette IN

Investigators

Abstract

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.