CAREER: Automorphic Forms and the Langlands Program

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 integer solutions to polynomial equations) and analysis (e.g., automorphic forms, which are highly symmetric solutions to certain differential equations on symmetric manifolds). 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. The project also integrates educational opportunities, including public outreach lectures, undergraduate and graduate research activities, cross-disciplinary training and research, and graduate curriculum development. A main theme in the theory of automorphic forms is to study the discrete spectrum of a connected reductive algebraic group defined over a number field. By the pioneering work of Arthur, followed by many others, the discrete spectrum of a classical group has been classified into so-called Arthur packets, which are 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 PI will also work on establishing Langlands functorial descent for exceptional groups, towards studying the Langlands functoriality and the discrete spectra of exceptional groups. 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, as well as converse problems for exceptional groups. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.