Iwasawa theory for cyclotomic towers

泰勒-怀尔斯系统的概念是通过分析椭圆曲线上谷山-志村猜想的部分答案而产生的。这种新的公理化方法和欧拉系统正在成为岩川理论的基本工具。在此研究期间，Taylor-Wiles系统方法在以下几个方面得到了发展：a) Hida的近普通Hecke代数的R=T定理，b)全实场的环切塔的研究，c)高维酉Shimura变的Taylor-Wiles系统的构造。在a)中，证明了H.Hida （UCLA）定义的与剩余不可约表示相对应的几乎普通的Hecke代数在几乎所有情况下都被万向变形环识别。在b)中，我制定了Iwasawa-Greenberg猜想的非阿贝尔版本。为了研究这一问题，提出了一种分环塔的变形理论。对于特殊的二维表示，利用泰勒-威尔斯系统的技术，发现这个新问题等价于经典的Iwasawa-Greenberg猜想。这一结果于2000年4月在CEB（法国巴黎）举行的自同构形式国际会议上公布。在c)中，针对上同调群的正则积分结构构造了Taylor-Wiles系统。该结果已在“代数几何2000”国际研讨会（2000年7月，日本长野）、第三届亚洲数学家大会（2000年10月，菲律宾马尼拉）和“自同态形式和志村变体”国际研讨会（2001年3月，美国巴尔的摩）上公布。除了这些口头交流，这些结果以预印本形式分发，并提交给期刊。

The notion of Taylor-Wiles system was born by analyzing a partial anwer to the Taniyama-Shimura conjecture on elliptic curves. This new axiomatic approach as well as Euler system is becoming a basic tool in Iwasawa theory. During this research period, Taylor-Wiles system approaches have been developed in the following directions :a) R=T theorems for Hida's nearly ordinary Hecke algebras,b) Study of cyclotomic towers of totally real fields,c) Construction of Taylor-Wiles systems for higher dimensional unitary Shimura varieties.In a), it is shown that nearly ordinary Hecke algebras defined by H.Hida (UCLA) corresponding to residually irreducible representations are identified with universal deformation rings in almost all cases.In b), I have formulated a non-abelian version of Iwasawa-Greenberg conjecture.To study this problem, a deformation theory over cyclotomic tower is developed. For special 2-dimensional representations, it is found that this new problem is equivalent to the classical Iwasawa-Greenberg conjecture by the technique of Taylor-Wlles systems. This result was announced at the international conference on automorphic forms at CEB (Paris, France) in April 2000.In c), Taylor-Wiles systems are constructed for the canonical integral structure of the cohomology groups. The result is announced at the international workshop "Algebraic Geometry 2000" (July 2000, Nagano, Japan), the third Asian Congress of Mathematicians (Oct. 2000, Manila, Philippine), and the international workshop "Automorphic forms and Shimura varieties" (March 2001, Baltimore, USA).Besides these oral communications, these results are distributed in a preprint form, and submitted to Journals.