Automorphic Forms on Shimura Varieties and L-functions

DMS-0244401飞，春三摘要：主要研究者和他的合作者将着手研究志村品种的辛和酉型算术几何。 在发展了还原群上的自守形式的p-adicdeformation理论之后，人们现在应该能够理解它在代数数论中更多的算术研究中的全部含义。 我们追求以下目标：1.一般线性群的p-adic自守L-函数的构造;2.证明非零模一个给定的素数p这样的L-值;3.确定L-函数的p整除性;4.在非阿贝尔环境下反分圆主图的可能证明。除了这些主要项目外，研究者将与他的研究生一起研究，本文讨论了一般全真实的域上的Hilbert第12问题和由不可解模p Galois表示构成的Shimura变分zeta函数.系统地研究了系数为（p-adic）的椭圆模形式解析函数是由主要研究者开始的，并已发展成为较大经典群上自守形式的变形理论。椭圆理论有许多深刻的应用。例如，它被怀尔斯用作证明岩泽猜想的重要组成部分，被怀尔斯和R. Taylor和R.泰勒和他的合作者。 主要作者将追求这样的应用在一个更一般的框架自守形式的经典群体。 这将导致非阿贝尔类数公式的新情况（即，自守岩泽理论的一些主要假设的证明），这些公式将纯粹算术定义的不变量与纯粹解析定义的zeta函数值联系起来（从而开辟了一种通过解析方法计算这些数字的方法）。

DMS-0244401Hida, HaruzoAbstract:The principal investigator and his collaboratorswill embark on his study of arithmetic geometry of Shimura varieties of symplectic and unitary type. After the development of the p-adicdeformation theory of automorphic forms on reductive groupsadmitting Shimura varieties, one should now be able to fathom its full implication in more arithmetic research in algebraicnumber theory. We pursue the following goals:1. construction of p-adic automorphic L-functions of generallinear 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 conjecturesin an non-abelian setting.In addition to these main projects, the investigator will studyjointly with his graduate students, the zeta function of Shimura varietiestwisted 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 coefficientsare (p-adic) analytic functions was startedby 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-Taniyamaconjecture by Wiles and R. Taylor and of a proof of theArtin conjecture of many non-soluble two dimensional Artin representationsby R. Taylor and his collaborators. The principal investigatorwill pursue such applications in a more general frameworkof 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 analyticallydefined values of zeta functions (thus opening a way of computingsuch numbers by analytic means).