L- functions, Galois representations and their arithmetic

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

The earlier and present work of the principal investigator (PI) has been and continues to be influential upon researchers world-wide. His work produced many new results on the following three topics: (a) Galois deformation rings = Hecke algebras (the "R=T" theorems); (b) p-adic analytic families of automorphic forms and their L-functions; (c) analysis of arithmetic invariants of p-adic L-functions. The PI's work has proven intellectual merit. The study of classical and p-adic modular forms has seen explosive development since the 1995 proof by Wiles and Taylor of cases of the Shimura-Taniyama Conjecture (via (a)), and hence, by a celebrated 1986 work of Ribet, Fermat's Last Theorem. Consequent developments (by the collaboration of many number theorists) include the complete proof of the above conjecture in all cases in 1998, the proof of the Local Langlands Conjecture for GL(N) in 1999, the proof of most instances of the icosahedral case of the Artin Conjecture, and the extension in numerous works of the p-adic theory of Wiles and Taylor from GL(2) to other reductive groups, especially to unitary groups. Despite this great and rapid development, several key results seemed quite out of reach, notably the Serre Modularity Conjecture (1986), the Fontaine-Mazur Conjecture (1990), and the Sato-Tate Conjecture (1965). It was therefore all the more surprising when in 2005--2006 all three of these conjectures were proven. All these works are directly related to the works of Hida/Tilouine and have good component under the influence of the PI's (present/past) work. The PI and his collaborator J. Tilouine will continue to work out new problems in these three areas of research, and their research influence upon younger generation will continue to be strong. The proposed work has had and will have broader impacts. The invited talks in the number theory section at ICM (Madrid in 2006) were dominantly on these topics; for example, Fujiwara described his results on (a); Skinner-Urban described their results on (b) and Vatsal touched on the topic (c). In their reports in the ICM proceedings, the PI's contribution/impact is well documented. Since the PI's five research/text books written in the past has had good audience and strong impact on the development of the theory, he plans to write two new books describing the topics in the items (b) and (c) so that graduate students and researchers will have good access to these cutting-edge and fast-developing research topics. The goal of the research under this plan is the development of an arithmetic theory of automorphic forms on Shimura varieties and their $L$-functions and Galois representations. Here is a list of the projects as a summary of the program: I. p-Adic automorphic measure; II. Iwasawa's mu-invariant; III. Non-vanishing modulo p of L-values; IV. Anticyclotomic main conjectures for CM fields; V. L-invariant of Tate curves; VI. L-invariant of abelian p-adic L-functions; VII. Base-change and L-invariant; VIII. Companion forms and local indecomposability.