On Square Integrable Automorphic Forms and Related Problems

Abstract for the proposal DMS-0653742 of Jiang

The Principal Investigator, Dihua Jiang, plans to continue his research on automorphic forms, L-functions and applications to number theory and arithmetic. In general, the PI aims to investigate basic structures of automorphic forms by using harmonic analysis and groups representations. In particular, the PI attacks basic conjectures of R. Langlands in the modern theory of automorphic forms. The proposed research of the PI leads to better understanding the discrete spectrum of the space of square integrable automorphic forms as conjectured by J. Arthur, the related problems on the central values of automorphic L-functions and the corresponding problems in the local theory. Applications to arithmetic of Shimura varieties and to basic problems on Galois groups have been obtained or are expected to be obtained in near future.

Basic objects in the universe may be classified by their intrinsic patterns. The basic types of these intrinsic patterns are called types of symmetry, which may be classified mathematically by group representations. Harmonic analysis provides mathematically methods to investigate the symmetric structures via functions. Automorphic functions are special functions with abundant symmetric structures and have natural connections to geometry and number theory. As mathematical subject, automorphic functions have been a very active research area for centuries and will continue to be so for centuries to come. The Principal Investigator, Dihua Jiang, proposes to investigate basic symmetric structures of automorphic functions and their applications to important problems in number theory and arithmetic.