Automorphic Forms on Reductive Groups and Their Covers

Boston College, Chestnut Hill MA

Investigators

Abstract

This research project studies functions arising in number theory that exhibit special symmetry properties under transformations, called automorphic forms. Some of these functions have coefficients that encode important information in arithmetic, and some have recently been connected to aspects of string theory in physics. The fundamental Langlands Functoriality Conjectures predict subtle relations between different spaces of automorphic forms, a structure that is closely related to many questions in number theory and analysis. This research project focuses on establishing properties of automorphic forms and new connections between different spaces of automorphic forms. The project will also support a graduate student and allow the PI and his students to disseminate the work through conferences and seminars. This project treats automorphic forms and representations on reductive groups and on their metaplectic covers of arbitrary degree. Automorphic forms on reductive groups are the key ingredients of the Langlands Program, and automorphic forms on covers are closely tied to reciprocity laws and related to the congruence subgroup problem. The research focuses on correspondences and on L-functions. In one series of projects, the principal investigator will develop and study new theta correspondences that generalize the classical theta correspondence but involve the tensor product of two small representations. A second project seeks to give a new Shimura correspondence that is detected by a period involving a theta function on an orthogonal group. This will be established by means of a new relative trace formula. A third set of projects concerns the systematic development of integral representations for L-functions which make use of the residual spectrum. These projects will advance our knowledge of automorphic forms, both on reductive groups and on finite degree covers, our understanding of small automorphic representations, and our understanding of the relations between automorphic forms on different groups. It will advance our knowledge of number theory and of representation theory. 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.