L- functions, Galois representations and their arithmetic

主要研究者(PI)的早期和现在的工作一直并继续对世界各地的研究人员产生影响。他的工作在以下三个主题上产生了许多新的结果：(A)伽罗华形变环=Hecke代数(“R=T”定理)；(B)p-进的自同构形式及其L函数的解析族；(C)p-进的L函数的算术不变量的分析。自1995年Wiles和Taylor证明Shimura-Taniyama猜想的情形(通过(A))以来，经典和p-进模形式的研究有了爆炸性的发展。因此，由Ribet 1986年的著名著作费马最后定理。随后的发展(通过许多数学家的合作)包括1998年在所有情况下对上述猜想的完整证明，1999年对GL(N)的局部朗兰兹猜想的证明，Artin猜想的二十面体情况的大多数情况的证明，以及在许多著作中Wiles和Taylor的p-进理论从GL(2)推广到其他约化群，特别是到酉群。尽管有了如此巨大和迅速的发展，但有几个关键结果似乎是遥不可及的，特别是Serre模数猜想(1986)、方丹-马祖尔猜想(1990)和佐藤泰特猜想(1965)。因此，当2005-2006年所有这三个猜想都得到证实时，这就更加令人惊讶。所有这些作品都与伊达/提鲁因的作品直接相关，并在国际派(现在/过去)作品的影响下具有良好的组成部分。PI和他的合作者J.Tilouine将继续解决这三个研究领域的新问题，他们对年轻一代的研究影响将继续强大。拟议的工作已经并将产生更广泛的影响。ICM(2006年马德里)数论部分的特邀演讲主要涉及这些主题；例如，藤原在(A)上描述了他的结果；斯金纳-厄本在(B)上描述了他们的结果，Vatsal在(C)上触及了这个主题。他计划写两本新书来描述(B)和(C)项中的主题，以便研究生和研究人员能够很好地接触到这些前沿和快速发展的研究主题。在这项计划下的研究目标是发展关于Shimura变种及其$L$-函数和伽罗瓦表示的自同构形式的算术理论。以下是作为项目总结的项目列表：i.p-adic自同构测度；ii.岩泽的u-不变量；iii.L值的非零模p；iv.CM域的反闭合主要猜想；V.L-泰特曲线不变量；vi.交换p元L-函数的L不变量；vii.基变化与L不变量；viii.伴随形式与局部不可分解.

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.