Topics in the Theory of Automorphic Representations

University Of Minnesota-Twin Cities, Minneapolis MN

Investigators

Abstract

The investigator studies two fundamental problems in the modern theory of automorphic forms. The first problem is concerned with the intrinsic symmetric structures of automorphic forms of reductive groups over number fields. Such structures of automorphic forms can be characterized by the existence of various period integrals of automorphic forms. Periods become recently a very useful tool to characterize various basic properties of automorphic forms and L-functions. In this project, the investigator and his colleagues intend to use periods to characterize new families of automorphic forms, especially, the degenerate cuspidal automorphic forms. The first example of such degenerate cuspidal automorphic forms is systematically studied by I. Piatetski-Shapiro in early 1980's in order to understand the generalization of the Ramanujan conjecture on estimate of Fourier coefficients of automorphic forms. These new families of automorphic forms characterized by the relevant periods are expected to share the basic structures with the ones suggested by J. Arthur from his work on general trace formula over the space of square integrable automorphic forms of reductive groups. The second problem is to study local and global Langlands reciprocity laws. The investigator and his colleague intend to establish local Langlands reciprocity law for classical groups by studying local gamma functions and using recent progress in the local and global theory of automorphic forms. The investigator and his colleagues study the theory of automorphic forms, an abstract, transcendental theory that describes repeating phenomena like sound and waves. A cornerstone of modern mathematics, it provides the foundation for relevant areas of mathematics and has applications to computer network theory and string theory in physics. Using this theory, the investigator and his colleagues can examine the reciprocal relations among objects from geometry, analysis, and number theory. The theory played an essential role in recent solution of the Fermat's last theorem by A. Wiles, and has also been important to the recent development in coding theory and cryptology. The investigator and his collaborators study in the project the intrinsic symmetric structures and transcendental invariants of automorphic forms.