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New Development of Iwasawa theory

Research Project

Project/Area Number 19H01783
Research Category

Grant-in-Aid for Scientific Research (B)

Allocation TypeSingle-year Grants
Section一般
Review Section Basic Section 11010:Algebra-related
Research InstitutionKeio University

Principal Investigator

Kurihara Masato  慶應義塾大学, 理工学部(矢上), 教授 (40211221)

Co-Investigator(Kenkyū-buntansha) 塩川 宇賢  慶應義塾大学, 理工学部(矢上), 名誉教授 (00015835)
池田 保  京都大学, 理学研究科, 教授 (20211716)
藤井 俊  島根大学, 学術研究院教育学系, 准教授 (20386618)
Project Period (FY) 2019-04-01 – 2023-03-31
Project Status Completed (Fiscal Year 2022)
Budget Amount *help
¥16,900,000 (Direct Cost: ¥13,000,000、Indirect Cost: ¥3,900,000)
Fiscal Year 2021: ¥2,730,000 (Direct Cost: ¥2,100,000、Indirect Cost: ¥630,000)
Fiscal Year 2020: ¥6,760,000 (Direct Cost: ¥5,200,000、Indirect Cost: ¥1,560,000)
Fiscal Year 2019: ¥5,330,000 (Direct Cost: ¥4,100,000、Indirect Cost: ¥1,230,000)
KeywordsIwasawa theory / zeta elements / Iwasawa module / Fitting ideal / 楕円曲線 / Selmer群 / Gauss和型Euler系 / 岩澤理論 / 保形形式 / イデアル類群 / Fittingイデアル / zeta元 / Mazur Tate予想 / 整数論 / 同変岩澤理論 / ゼータ元
Outline of Research at the Start

岩澤理論の最近の発展に伴う多くの研究を行うが、その中でも重要なのは、古典的岩澤加群を同変的に研究し、p 進 L 関数との精密な関係を確立すること、次にKatoのzeta elementについての新しい性質を述べた予想を定式化し、他の有名な予想との関係を確立すること、さらにはSelmer群の構造とmodular symbolについて、今まで研究代表者が得ていた結果をさらに発展させることである。

Outline of Final Research Achievements

We discovered a new property on Beilinson-Kato elements for elliptic curves, using Darmon-type derivatives. We call this property Generalized Perrin-Riou Conjecture, and studied it in detail. In particular, we showed that it implies the famous Mazur Tate conjecture, a refinement of the Birch and Swinnerton-Dyer conjecture, under certain conditions. This is joint work with David Burns and Takamichi Sano.
We also compute the Fitting ideal of the classical Iwasawa module over the cyclotomic Zp-extension of a totally real field. This is joint work with Cornelius Greither and Takenori Kataoka.

Academic Significance and Societal Importance of the Research Achievements

有理数体上の楕円曲線の数論は、さまざまな理論への一般化が可能であり、またさまざまな分野への応用も可能である。そこで、この分野で新しい性質を見出すことは、きわめて大きな価値がある。また、岩澤理論をFittingイデアルを用いて定式化し直すことは、その精密化を得ることにもなり、大変価値のあることである。古典的岩澤加群を扱うことは、扱いやすいコホモロジー群を扱うよりも難しく、このような加群に対して理論を構築することの学術的意義は高い。

Report

(4 results)
  • 2022 Final Research Report ( PDF )
  • 2021 Annual Research Report
  • 2020 Annual Research Report
  • 2019 Annual Research Report
  • Research Products

    (23 results)

All 2022 2021 2020 2019 Other

All Int'l Joint Research (7 results) Journal Article (8 results) (of which Int'l Joint Research: 6 results,  Peer Reviewed: 8 results,  Open Access: 2 results) Presentation (8 results) (of which Int'l Joint Research: 3 results,  Invited: 7 results)

  • [Int'l Joint Research] King's College London(英国)

    • Related Report
      2021 Annual Research Report
  • [Int'l Joint Research] ミュンヘン防衛大(ドイツ)

    • Related Report
      2021 Annual Research Report
  • [Int'l Joint Research] King's College London(英国)

    • Related Report
      2020 Annual Research Report
  • [Int'l Joint Research] ミュンヘン防衛大(ドイツ)

    • Related Report
      2020 Annual Research Report
  • [Int'l Joint Research] King's College London(英国)

    • Related Report
      2019 Annual Research Report
  • [Int'l Joint Research] Universitaet der Bundeswehr Muenchen(ドイツ)

    • Related Report
      2019 Annual Research Report
  • [Int'l Joint Research] Korea Institute for Advanced Study(韓国)

    • Related Report
      2019 Annual Research Report
  • [Journal Article] Notes on the dual of the ideal class groups of CM-fields2021

    • Author(s)
      Masato Kurihara
    • Journal Title

      Journal de Theorie des Nombres de Bordeaux

      Volume: 33 Issue: 3.2 Pages: 971-996

    • DOI

      10.5802/jtnb.1184

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] Fitting ideals of p-ramified Iwasawa modules over totally real fields2021

    • Author(s)
      Greither Cornelius, Kataoka Takenori, Kurihara Masato
    • Journal Title

      Selecta Mathematica

      Volume: 28 Issue: 1 Pages: 1-48

    • DOI

      10.1007/s00029-021-00731-5

    • Related Report
      2021 Annual Research Report
    • Peer Reviewed / Open Access / Int'l Joint Research
  • [Journal Article] On Stark elements of arbitrary weight and their p-adic families2020

    • Author(s)
      David Burns, Masato Kurihara, Takamichi Sano
    • Journal Title

      Advanced Studies in Pure Mathematics

      Volume: 86 Pages: 113-140

    • DOI

      10.2969/aspm/08610113

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] The second syzygy of the trivial G-module, and an equivariant main conjecture2020

    • Author(s)
      Cornelius Greither, Masato Kurihara and Hibiki Tokio
    • Journal Title

      Development of Iwasawa Theory- the Centennial of K. Iwasawa's Birth, Advanced Studies in Pure Mathematics

      Volume: 86 Pages: 317-349

    • DOI

      10.2969/aspm/08610317

    • Related Report
      2020 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] On the lifting of Hilbert cusp forms to Hilbert-Siegel cusp forms2020

    • Author(s)
      T. Ikeda and S. Yamana
    • Journal Title

      Ann. Sci. Ec. Norm. Sup. (4)

      Volume: -

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On Stark elements of arbitrary weight and their p-adic families2020

    • Author(s)
      David Burns, Masato Kurihara and Takamichi Sano
    • Journal Title

      Proceedings of Iwasawa 2017, Advanced Studies in Pure Mathematics

      Volume: -

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] The second syzygy of the trivial G-module, and an equivariant main conjecture2020

    • Author(s)
      Cornelius Greither, Masato Kurihara and Hibiki Tokio
    • Journal Title

      Proceedings of Iwasawa 2017, Advanced Studies in Pure Mathematics

      Volume: -

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] On the Refined Conjectures on Fitting Ideals of Selmer Groups of Elliptic Curves with Supersingular Reduction2019

    • Author(s)
      Kim Chan-Ho, Kurihara Masato
    • Journal Title

      International Mathematics Research Notices

      Volume: - Issue: 14 Pages: 10559-10599

    • DOI

      10.1093/imrn/rnz129

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed / Int'l Joint Research
  • [Presentation] Dasgupta Kakde の最近の仕事とその周辺 Overview2022

    • Author(s)
      栗原 将人
    • Organizer
      Dasgupta Kakde の最近の仕事とその周辺
    • Related Report
      2021 Annual Research Report
    • Invited
  • [Presentation] いくつかの Selmer module の導入とその性質2022

    • Author(s)
      栗原 将人
    • Organizer
      Dasgupta Kakde の最近の仕事とその周辺
    • Related Report
      2021 Annual Research Report
    • Invited
  • [Presentation] Some analytic quantities in the arithmetic of elliptic curves2021

    • Author(s)
      Masato Kurihara
    • Organizer
      The 9th East Asia Number Theory Conference
    • Related Report
      2021 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Survey on the Brumer-Stark conjecture and the proof by Dasgupta and Kakde2020

    • Author(s)
      栗原将人
    • Organizer
      代数的整数論とその周辺
    • Related Report
      2020 Annual Research Report
  • [Presentation] Derivatives of Kato's Euler system for elliptic curves2020

    • Author(s)
      Masato Kurihara
    • Organizer
      International Colloquium on Arithmetic Geometry(Tata Institute of Fundamental Research, Mumbai)
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Several regulators and a new conjecture on Kato's zeta elements for elliptic curves2019

    • Author(s)
      Masato Kurihara
    • Organizer
      Regulators in Niseko 2019
    • Related Report
      2019 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 円分体のイデアル類群のマイナスパートの挙動について2019

    • Author(s)
      藤井俊
    • Organizer
      愛知数論セミナー
    • Related Report
      2019 Annual Research Report
    • Invited
  • [Presentation] 円分体のイデアル類群のマイナスパートの挙動について2019

    • Author(s)
      藤井俊
    • Organizer
      北陸数論セミナー
    • Related Report
      2019 Annual Research Report
    • Invited

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Published: 2019-04-18   Modified: 2024-01-30  

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