On the Theory of Automorphic Forms and Applications

University Of Minnesota-Twin Cities, Minneapolis MN

Investigators

Abstract

Abstract of Dihua Jiang's proposal (DMS-0400414) The Principal Investigator, Dihua Jiang, works on some basic problems in the modern theory of automorphic forms and L-functions. In the global theory, he investigates the structures of the discrete spectrum of automorphic forms and characterizes the non-vanishing of the central values or the residues of automorphic L-functions in terms of periods of automorphic forms, using ideas from the classical invariant theory. In the local theory, his research attacks the local Langlands conjectures and related basic problems in harmonic analysis of p-adic groups. His long term goal is to understand the general local-global-automorphic principles in the theory of automorphic forms, which reflects one of the basic principles in arithmetic and number theory. The theory of automorphic forms has been studied for a long time and was rooted in the work of Gauss and Riemann among others. It became early on a meeting ground for analysis and number theory, and has developed as an indispensable tool in analytic number theory, algebraic number theory, Diophantine problems, arithmetic and algebraic geometry, and recently in infinite dimensional Lie algebras and mathematical physics. The recent proof of the Taniyama-Shimura-Weil conjecture on elliptic curves and the Fermat's last theorem are typical examples. The modern theory of automorphic forms and L-functions was developed based on the understanding of automorphic forms in terms of the representation theory of and the harmonic analysis over locally compact topological groups. Langlands gave a systematic conjectural description of relations between L-functions in number theory or algebraic geometry and those arising in the theory of automorphic forms. Jiang's work is concentrated on the basic conjectures of Langlands and applications to number thoery.