Iwasawa theory for cyclotomic towers

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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.