Automorphic Forms on Shimura Varieties and L-functions

DMS-0244401 Hida，Haruzo摘要：主要研究人员和他的合作者将着手研究Shimura辛型和酉型变种的算术几何。在允许Shimura簇的约化群上发展了自同构形的p-Add形变理论之后，现在应该能够在更多的代数数论的算术研究中理解它的全部含义。我们的目标是：1.构造一般线性群的p-进自同构的L-函数；2.证明这类L值的模a不为零；3.确定L-函数的p的可除性；4.非交换环境下反闭合主要猜想的可能证明。除了这些主要项目外，研究人员还将与他的研究生一起研究由非可解模伽罗瓦表示产生的Shimura族的Zeta函数和Hilbert在一般全实域上的第十二个问题。主要研究人员开始了对其系数为(p-进)解析函数的椭圆模形式的系统研究，并已发展成更大经典群上的自同构形的形变理论。椭圆理论得到了无数深刻的应用。例如，它被用作Wiles对岩泽猜想的证明，Wiles和R.Taylor对长期存在的Fermat最后定理和Shimura-Taniyamaconect的证明，以及R.Taylor和他的合作者对许多不可解的二维Artin表示的Artin猜想的证明。主要研究人员将在经典群上的自同构形式的更一般的框架中探索这样的应用。这将导致非阿贝尔类数公式的新情况(即，对自同构岩泽理论的一些主要猜想的证明)，这些公式将纯算术定义的不变量与纯解析定义的Zeta函数的值联系起来(从而开辟了一种用解析方法计算此类数的方法)。

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).