Iwasawa theory for cyclotomic towers

Taylor-Wiles系统的概念是通过对椭圆曲线上的Taniyama-Shimura猜想的部分解答的分析而产生的。这种新的公理化方法和欧拉系统一样，正在成为岩泽理论的基本工具。在此期间，Taylor-Wiles系统方法在以下几个方面得到了发展：a）希达的近似普通Hecke代数的R=T定理，b）全真实的域的分圆塔的研究，c）高维酉Shimura簇的Taylor-Wiles系统的构造。在a）中，证明了由H.希达（UCLA）定义的近似普通Hecke代数对应于剩余不可约表示在几乎所有情况下都与泛变形环相同。在B）中，我提出了Iwasawa-Greenberg猜想的一个非交换版本。为了研究这个问题，发展了分圆塔上的变形理论。对于特殊的2维表示，利用Taylor-Wlles系统的技巧，发现这个新问题等价于经典的Iwasawa-Greenberg猜想.这一结果在2000年4月于法国巴黎举行的国际自守形式会议上公布。在c）中，对上同调群的典型整结构构造了Taylor-Wiles系统。这一结果在2000年国际代数几何研讨会上公布第三届亚洲数学家大会（2000年7月，日本长野）（2000年10月，菲律宾马尼拉），以及“自守形式和志村品种”国际讲习班（2001年3月，巴尔的摩，美国）。除了这些口头交流，这些结果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.