Automorphic Forms on Shimura Varieties and L-functions

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

DMS-0244401 Hida, Haruzo Abstract: The principal investigator and his collaborators will embark on his study of arithmetic geometry of Shimura varieties of symplectic and unitary type. After the development of the p-adic deformation theory of automorphic forms on reductive groups admitting Shimura varieties, one should now be able to fathom its full implication in more arithmetic research in algebraic number theory. We pursue the following goals: 1. construction of p-adic automorphic L-functions of general linear groups; 2. proof of non-vanishing modulo a given prime p of such L-values; 3. determination of divisibility by p of the L-functions; 4. possible proof of the anticyclotomic main conjectures in an non-abelian setting. In addition to these main projects, the investigator will study jointly with his graduate students, the zeta function of Shimura varieties twisted by non-soluble mod p Galois representations and Hilbert's twelfth problem over general totally real fields. A systematic study of elliptic modular forms whose coefficients are (p-adic) analytic functions was started by the principal investigator and has been developed into a deformation theory of automorphic forms on larger classical groups. The elliptic theory has found numerous profound applications. For example, it was used as an essential ingredient of a proof of Iwasawa's conjecture by Wiles, of the proof of longstanding Fermat's last theorem and the Shimura-Taniyama conjecture by Wiles and R. Taylor and of a proof of the Artin conjecture of many non-soluble two dimensional Artin representations by R. Taylor and his collaborators. The principal investigator will pursue such applications in a more general framework of automorphic forms on classical groups. This would results new cases of non-abelian class number formulas (that is, a proof of some main conjectures of automorphic Iwasawa theory) which connect purely arithmetically defined invariants to purely analytically defined values of zeta functions (thus opening a way of computing such numbers by analytic means).