Project/Area Number  10640018 
Research Category 
GrantinAid for Scientific Research (C).

Section  一般 
Research Field 
Algebra

Research Institution  Nagoya University 
Principal Investigator 
FUJIWARA Kazuhiro Nagoya univ., graduate school of math., ass.prof., 大学院・多元数理科学研究科, 助教授 (00229064)

CoInvestigator(Kenkyūbuntansha) 
OCHIAI Hiroyuki Kyushu Univ., graduate school of math.sci., ass.prof., 大学院・数理学研究科, 助教授 (90214163)
UZAWA Tohru Rikkyo Univ., dept.of math., ass.prof., 理学部, 助教授 (40232813)
MUKAI Shigeru Nagoya Univ., graduate school of math., prof., 大学院・多元数理科学研究科, 教授 (80115641)
SAITO Shuji Nagoya Univ., graduate school of math., prof., 大学院・多元数理科学研究科, 教授 (50153804)
SAITO Takeshi Univ.of Tokyo, graduate school of math.sci., prof., 大学院・数理科学研究科, 教授 (70201506)

Project Fiscal Year 
1998 – 2000

Project Status 
Completed(Fiscal Year 2000)

Budget Amount *help 
¥3,300,000 (Direct Cost : ¥3,300,000)
Fiscal Year 2000 : ¥800,000 (Direct Cost : ¥800,000)
Fiscal Year 1999 : ¥1,100,000 (Direct Cost : ¥1,100,000)
Fiscal Year 1998 : ¥1,400,000 (Direct Cost : ¥1,400,000)

Keywords  Galois representation / noncommutative class field theory / automorphic form / automorphic representation / Iwasawa theory / ガロア表現 / 非可換類体論 / 保型形式 / 保型表現 / 岩沢理論 / 岩澤理論 / 岩澤Greenberg 予想 / 円分体 / ヘッケ環 / 岩沢加群 / 岩沢主予想 
Research Abstract 
The notion of TaylorWiles system was born by analyzing a partial anwer to the TaniyamaShimura 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, TaylorWiles 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 TaylorWiles 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 nonabelian version of IwasawaGreenberg conjecture. To study this problem, a deformation theory over cyclotomic tower is developed. For special 2dimensional representations, it is found that this new problem is equivalent to the classical IwasawaGreenberg conjecture by the technique of TaylorWlles systems. This result was announced at the international conference on automorphic forms at CEB (Paris, France) in April 2000. In c), TaylorWiles 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.
