Iwasawa theory for cyclotomic towers

Taylor-Wiles系统的概念是通过分析椭圆曲线上的Taniyama-Shimura猜想的一个偏解而产生的。这种新的公理方法和欧拉系统一样，正在成为岩泽理论的基本工具。在这一研究期间，Taylor-Wiles系统的方法在以下方面得到了发展：a)Hida的近普通Hecke代数的R=T定理，b)全实域的分圆塔的研究，c)高维酉Shimura变种的Taylor-Wiles系统的构造。在a)，证明了几乎在所有情况下，由H.Hida(UCLA)定义的对应于剩余不可约表示的几乎普通的Hecke代数都与泛变形环一致。在b)，我建立了一个非交换版本的岩泽-Greenberg猜想。对于特殊的二维表示，利用Taylor-Wlles系统的技巧，我们发现这个新问题等价于经典的岩泽-格林伯格猜想。这一结果是在2000年4月法国巴黎CEB国际会议上宣布的。在c)中，上同调群的标准积分结构被构造为Taylor-Wiles系统。这一结果是在国际研讨会“代数几何2000”(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)和国际研讨会“自同构形式和下村变种”(2001年3月，美国巴尔的摩)上公布的。除了这些口头交流，这些结果还以预印本的形式分发，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.