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Iwasawa theory for cyclotomic towers

Research Project

Project/Area Number 10640018
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionNagoya 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)

Co-Investigator(Kenkyū-buntansha) SAITO Shuji  Nagoya Univ., graduate school of math., prof., 大学院・多元数理科学研究科, 教授 (50153804)
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 Takeshi  Univ.of Tokyo, graduate school of math.sci., prof., 大学院・数理科学研究科, 教授 (70201506)
Project Period (FY) 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)
KeywordsGalois representation / non-commutative class field theory / automorphic form / automorphic representation / Iwasawa theory / 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 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.

Report

(4 results)
  • 2000 Annual Research Report   Final Research Report Summary
  • 1999 Annual Research Report
  • 1998 Annual Research Report
  • Research Products

    (23 results)

All Other

All Publications (23 results)

  • [Publications] K.Fujiwara: "Rigid geometry and etale cohomology of schemes"京都大学数理解析研究所講究録. 1073. 168-190 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] K.Fujiwara: "A proof of flattening theorem in the formal case"Nagoya Journal of Math.. (印刷中).

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Ochiai: "A p-adic property of the Taylor series of exp (x+x^p/p)"Hokkaido Math.J.. 28. 71-85 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Ochiai: "Bernstein degree of singular unitary highest weight representations of the metaplectic group (with K.Nishiyama)"Proc.Japan Acad.. 75. 9-11 (1999)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] T.Saito: "Inequality for conductor and differentials of a curve over a local field (with Q.Liu)"J.of Algebraic Geometry. 9. 409-424 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] T.Saito: "Weight-monodromy conjecture for l-adic representations associated to modular forms"The arithmetic and geometry of algebraic cycles. 427-431 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 向井茂: "モジュライ理論1"岩波書店. 220 (1998)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] 向井茂: "モジュライ理論2"岩波書店. 235 (2000)

    • Description
      「研究成果報告書概要(和文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] K.Fujiwara: "Rigid geometry and etale cohomology of schemes"RIMS reseach notes. 1073. 168-190 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] K.Fujiwara: "A proof of flattening theorem in the formal case"Nagoya J.of Math.. (in press).

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Ochiai: "A p-adic property of the Taylor series of exp (x+x^p/p)"Hokkaido Math.J.. 28. 71-85 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] H.Ochiai: "Bernstein degree of singular unitary highest weight representation of the metaplectic group (with K.Nishiyama)"Proc.Japan.Acad.. 75. 9-11 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] T.Saito: "Inequality for conductor and differentials of a curve over a locall fiels (with Q.Liu)"J.of Algebraic geometry. 9. 409-424 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] T.Saito: "Weight-monodromy conjecture for 1-adic representations associated to modular forms"The arithmetic and geometry of algebraic cycles. 427-431 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] S.Mukai: "Theory of moduli 1,2"Iwanami shoten (in Japanese). 455 (1998)

    • Description
      「研究成果報告書概要(欧文)」より
    • Related Report
      2000 Final Research Report Summary
  • [Publications] K.Fujiwara: "Rigid geometry and etale cohomology of schemes"京都大学数理解析研究所講究録. 1073. 168-190 (1999)

    • Related Report
      2000 Annual Research Report
  • [Publications] K.Fujiwara: "A proof of flattening theorem in the formal case"Nagoya Journal of Math.. (印刷中).

    • Related Report
      2000 Annual Research Report
  • [Publications] H.Ochiai: "A p-adic property of the Taylor series of exp(x+x^p/p)"Hokkaido Math.J.. 28. 71-85 (1999)

    • Related Report
      2000 Annual Research Report
  • [Publications] H.Ochiai: "Bernstein degree of singular unitary highest weight representations of the metaplectic group (with K.Nishiyama)"Proc.Japan Acad.. 75. 9-11 (1999)

    • Related Report
      2000 Annual Research Report
  • [Publications] T.Saito: "Inequality for conductor and differentials of a curve over a local field (with Q.Liu)"J.of Algebraic Geometry. 9. 409-424 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] T.Saito: "Weight-monodromy conjecture for l-adic representations associated to modular forms"The arithmetic and geometry of algebraic cycles. 427-431 (2000)

    • Related Report
      2000 Annual Research Report
  • [Publications] 向井茂: "モジュライ理論1"岩波書店. 220 (1998)

    • Related Report
      2000 Annual Research Report
  • [Publications] 向井茂: "モジュライ理論2"岩波書店. 235 (2000)

    • Related Report
      2000 Annual Research Report

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Published: 1998-04-01   Modified: 2016-04-21  

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