Arithmetic of Automorphic Forms on Reductive Groups

University Of California-Los Angeles, Los Angeles CA

Investigators

Abstract

While the investigator was pursuing the goals described in the grant proposal to NSF made and funded three years ago, he has come up with a few workable ideas which may be useful in verifying certain number of fundamental conjectures on arithmetic of automorphic forms. We now start an attack in order to make these partially valid speculations into a methodology based on a solid foundation. Project A describe a way to establish theory of p-adic families of automorphic forms defined on Shimura varieties, particularly those carrying a canonical fiber system of abelian varieties. After establishing such control theory of p-adic automorphic forms, construction of p-adic L-functions associated to each family of automorphic forms is the goal of Project B. There should be some applications of such p-adic L-functions and p-adic families to Iwasawa theoretic understanding of CM fields, to verification of automorphic functoriality (like base-change; Project C) and to modularity problems of abelian varieties (Project D). There is a possibility of extending such theory to algebraic groups not associated to Shimura varieties. In Project E, the case of general linear groups is discussed. If successful, many rationality results of automorphic L-functions are expected even in this non-holomorphic case. Number theoretic questions, though formulated often in an elementary way, could be astonishingly difficult to solve. The Fermat's last theorem (which was actually a conjecture made by Fermat in the seventeenth century) has been solved finally by Wiles in 1995, surviving for 350 years of serious attacks by the strongests of mathematicians. In its solution, arithmetic study of modular forms and automorphic forms played essential roles in many ways. The programs described here are intended to broaden the applicability of some of the tools used in this endeavor to more general setting, encompassing more geometric objects: those spaces classifying algebraic equations whose solution-set (so called "abelian varieties") having a canonical algebra (group) structure. Each abelian variety is somehow (conjecturally) related to a very specific function satisfying many symmetries (so it is called an "automorphic form"). The solution of such classification problems (so-called "modularity problem") in the simplest case of dimension one is a key ingredient of the above mentioned proof of Wiles. One of the goals described in this award is to generalize this "modularity" result to higher dimensional abelian varieties.