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Strategic Reseach using Eisensterin classes to prove Conjectures in Arithmetic Geometry

Research Project

Project/Area Number 26247004
Research Category

Grant-in-Aid for Scientific Research (A)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionKeio University

Principal Investigator

Bannai Kenichi  慶應義塾大学, 理工学部(矢上), 教授 (90343201)

Co-Investigator(Kenkyū-buntansha) 山本 修司  慶應義塾大学, 理工学研究科(矢上), 准教授 (20635370)
大坪 紀之  千葉大学, 大学院理学研究院, 教授 (60332566)
小林 真一  九州大学, 数理学研究院, 教授 (80362226)
安田 正大  大阪大学, 理学研究科, 准教授 (90346065)
Research Collaborator TAKAI Yuuki  
OTA Kazuto  
ONO Masataka  
KIRAL Erin Mehmet  
Project Period (FY) 2014-04-01 – 2019-03-31
Project Status Discontinued (Fiscal Year 2018)
Budget Amount *help
¥24,830,000 (Direct Cost: ¥19,100,000、Indirect Cost: ¥5,730,000)
Fiscal Year 2018: ¥4,940,000 (Direct Cost: ¥3,800,000、Indirect Cost: ¥1,140,000)
Fiscal Year 2017: ¥4,940,000 (Direct Cost: ¥3,800,000、Indirect Cost: ¥1,140,000)
Fiscal Year 2016: ¥4,940,000 (Direct Cost: ¥3,800,000、Indirect Cost: ¥1,140,000)
Fiscal Year 2015: ¥4,940,000 (Direct Cost: ¥3,800,000、Indirect Cost: ¥1,140,000)
Fiscal Year 2014: ¥5,070,000 (Direct Cost: ¥3,900,000、Indirect Cost: ¥1,170,000)
Keywords代数学 / 整数論 / 数論幾何 / L関数の特殊値 / ポリログ
Outline of Final Research Achievements

Our original goal was to study the polylogarithm in the case of totally real fields. Our original goal was to study the polylogarithm via the Eisenstein class, but in course of our research, we realized the importance of a certain algebraic torus associated to a totally real field, and using the ideas from plectic structures proposed by Nekovar and Scholl, we succeeded in proving that the Shintani generating function which generates special values of Shintani zeta functions, defines a canonical class on the algebraic torus.

Academic Significance and Societal Importance of the Research Achievements

整数論において、総実代数体は最も基本的な有理数体、虚2次体についで、最も基本的な研究対象である。また、有理数体と虚2次体の場合は対応する幾何学的対象は1次元であるが、総実代数体は高次元であることから、初めて遭遇する本質的に高次元な場合と言って良い。この場合に対して、70年代より新谷によりHecke L関数の特殊値を捉える新谷ゼータ関数とその母関数は研究されていたが、本研究では、この母関数をコホモロジー類で解釈すると、標準的な類によって与えられるという画期的な成果を得た。

Report

(5 results)
  • 2018 Final Research Report ( PDF )
  • 2017 Annual Research Report
  • 2016 Annual Research Report
  • 2015 Annual Research Report
  • 2014 Annual Research Report
  • Research Products

    (35 results)

All 2017 2016 2015 2014 Other

All Int'l Joint Research (2 results) Journal Article (6 results) (of which Peer Reviewed: 6 results,  Acknowledgement Compliant: 3 results,  Open Access: 2 results) Presentation (26 results) (of which Int'l Joint Research: 15 results,  Invited: 21 results) Funded Workshop (1 results)

  • [Int'l Joint Research] Regensburg大学(Germany)

    • Related Report
      2015 Annual Research Report
  • [Int'l Joint Research] Regensburg大学(Germany)

    • Related Report
      2014 Annual Research Report
  • [Journal Article] Finite multiple zeta values associated with 2-colored rooted trees2017

    • Author(s)
      Ono Masataka
    • Journal Title

      Journal of Number Theory

      Volume: 181 Pages: 99-116

    • DOI

      10.1016/j.jnt.2017.05.019

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Shuffle product of finite multiple polylogarithms2017

    • Author(s)
      Masataka Ono and Shuji Yamamoto
    • Journal Title

      Manuscripta Mathematica

      Volume: 152 Issue: 1-2 Pages: 153-166

    • DOI

      10.1007/s00229-016-0856-9

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] The radius of convergence of the p-adic sigma function2016

    • Author(s)
      Kenichi Bannai, Shinichi Kobayashi, Seidai Yasuda
    • Journal Title

      Mathematische Zeitschrift

      Volume: 印刷中 Issue: 1-2 Pages: 751-781

    • DOI

      10.1007/s00209-016-1783-x

    • Related Report
      2017 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Integral structures on pp-adic Fourier theory2016

    • Author(s)
      Kenichi Bannai and Shinichi Kobayashi
    • Journal Title

      Annales de L'Institut Fourier

      Volume: 66 Issue: 2 Pages: 521-550

    • DOI

      10.5802/aif.3018

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] $p$-adic Eisenstein-Kronecker series for CM elliptic curves and the Kronecker limit formulas2015

    • Author(s)
      Hidekazu Furusho, Kenichi Bannai and Shinichi Kobayashi
    • Journal Title

      Nagoya Math. J

      Volume: 219 Pages: 269-302

    • DOI

      10.1215/00277630-2891995

    • Related Report
      2015 Annual Research Report
    • Peer Reviewed / Open Access
  • [Journal Article] A sum formula of multiple L-values2015

    • Author(s)
      Shuji Yamamoto
    • Journal Title

      Int. J. Number Theory

      Volume: 11 Issue: 01 Pages: 127-137

    • DOI

      10.1142/s1793042115500074

    • Related Report
      2014 Annual Research Report
    • Peer Reviewed / Open Access
  • [Presentation] Category of mixed plectic Hodge structures2017

    • Author(s)
      K. Bannai, K. Hagihara, S. Kobayashi, K. Yamada, S. Yamamoto, S. Yasuda
    • Organizer
      Fukuoka International Conference on Arithmetic Geometry 2017
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Kawashima functions and multiple zeta values2017

    • Author(s)
      S. Yamamoto
    • Organizer
      Fukuoka International Conference on Arithmetic Geometry 2017
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] New functional equations of finite multiple polylogarithms2017

    • Author(s)
      小野雅隆
    • Organizer
      第11回福岡整数論研究集会
    • Related Report
      2017 Annual Research Report
  • [Presentation] プレクティック混合ホッジ構造について2017

    • Author(s)
      K. Bannai, K. Hagihara, S. Kobayashi, K. Yamada, S. Yamamoto, S. Yasuda
    • Organizer
      Regulators in Niseko 2017
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] 川島関数の諸公式と問題2017

    • Author(s)
      山本修司
    • Organizer
      第37回関西多重ゼータ研究会, 東北大学
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] New functional equations of finite multiple polylogarithms2017

    • Author(s)
      小野雅隆
    • Organizer
      第37回関西多重ゼータ研究会, 東北大学
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] New functional equations of finite multiple polylogarithms2017

    • Author(s)
      小野雅隆
    • Organizer
      Polylogs, multiple zetas, and related topics, Tohoku University
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On the Mazur-Tate refined conjecture for modular forms2017

    • Author(s)
      太田和惟
    • Organizer
      Workshop on p-adic L-functions and algebraic cycles, NTU Taipei
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On the Mazur-Tate refined conjecture for modular forms2017

    • Author(s)
      太田和惟
    • Organizer
      KIAS Number theory seminar 2017, KIAS Korea
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On the anticyclotomic Iwasawa main conjecture for modular forms2017

    • Author(s)
      太田和惟
    • Organizer
      UK-Japan Winter School 2018 on Number Theory, Kings College London
    • Related Report
      2017 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 2-1 公式とZhaoのbinomial identityに関する注意2017

    • Author(s)
      山本修司
    • Organizer
      第11回多重ゼータ研究集会&第39回関西多重ゼータ研究会
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] 多重ゼータ値の積分表示の拡張とその応用2017

    • Author(s)
      山本修司
    • Organizer
      代数的整数論とその周辺, 京都大学数理解析研究所
    • Related Report
      2017 Annual Research Report
  • [Presentation] Anticyclotomic Iwasawa main conjecture for modular forms2017

    • Author(s)
      太田和惟
    • Organizer
      代数的整数論とその周辺, 京都大学数理解析研究所
    • Related Report
      2017 Annual Research Report
  • [Presentation] Harmonic relation and integral expression for Kawashima functions2017

    • Author(s)
      山本修司
    • Organizer
      Arithmetic Geometry and Related Topics, 愛媛大学
    • Related Report
      2017 Annual Research Report
    • Invited
  • [Presentation] Multiple zeta functions associated with 2-colored rootedtrees2017

    • Author(s)
      小野雅隆
    • Organizer
      第11回ゼータ若手研究集会, 愛媛大学
    • Related Report
      2017 Annual Research Report
  • [Presentation] On values of the multiple zeta function of Kaneko-Tsumura type2016

    • Author(s)
      Shuji Yamamoto
    • Organizer
      Japan-Taiwan Joint conference on Number theory 2016
    • Place of Presentation
      Taipei (Taiwan)
    • Year and Date
      2016-09-10
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Finite multiple zeta values associated with 2-colored rooted trees2016

    • Author(s)
      Masataka Ono
    • Organizer
      Various Aspects of Multiple Zeta Values 2016
    • Place of Presentation
      京都大学数理解析研究所(京都府, 京都市)
    • Year and Date
      2016-07-12
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Integral-series identity, regularized double shuffle relation and Kawashima's relation2016

    • Author(s)
      Shuji Yamamoto
    • Organizer
      Various Aspects of Multiple Zeta Values 2016
    • Place of Presentation
      京都大学数理解析研究所(京都府, 京都市)
    • Year and Date
      2016-07-11
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On Kawashima's relation among multiple zeta values2016

    • Author(s)
      Shuji Yamamoto
    • Organizer
      Analogies between number fields and function fields: algebraic and analytic aspects
    • Place of Presentation
      Lyon (France)
    • Year and Date
      2016-06-28
    • Related Report
      2016 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Shuffle products of finite multiple polylogarithms2016

    • Author(s)
      M. Ono and S. Yamamoto
    • Organizer
      Workshop on Polylogarithms, MZVs and Mahler measures
    • Place of Presentation
      東北大学(宮城県・仙台市)
    • Year and Date
      2016-03-29
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] Kato's Euler system and the Mazur-Tate refined conjecture of BSD type2016

    • Author(s)
      K. Ota
    • Organizer
      Sendai International Conference on Arithmetic Geometry 2016
    • Place of Presentation
      東北大学(宮城県・仙台市)
    • Year and Date
      2016-01-28
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 加藤オイラー系とメイザー・テイトの BSD 型精密化予想について2015

    • Author(s)
      K. Ota
    • Organizer
      代数的整数論とその周辺2015
    • Place of Presentation
      京都大学数理解析研究所(京都府・京都市)
    • Year and Date
      2015-11-30
    • Related Report
      2015 Annual Research Report
  • [Presentation] The Kaneko-Tsumura multiple zeta functions of r+r variables2015

    • Author(s)
      S. Yamamoto
    • Organizer
      Zeta Functions of Several Variables and Applications
    • Place of Presentation
      名古屋大学(名古屋市・千種区)
    • Year and Date
      2015-11-13
    • Related Report
      2015 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] On Kawashima’s functions2014

    • Author(s)
      S. Yamamoto
    • Organizer
      Japan-Korea Joint Seminar on Number Theory and Related Topics
    • Place of Presentation
      慶應義塾大学(神奈川県・横浜市)
    • Year and Date
      2014-11-21
    • Related Report
      2014 Annual Research Report
    • Int'l Joint Research / Invited
  • [Presentation] 等号付き多重ゼータ値と2-1公式2014

    • Author(s)
      S. Yamamoto
    • Organizer
      第59回代数学シンポジウム
    • Place of Presentation
      東京大学(東京都・目黒区)
    • Year and Date
      2014-09-10
    • Related Report
      2014 Annual Research Report
    • Invited
  • [Presentation] On integral expressions of multiple zeta-star values2014

    • Author(s)
      S. Yamamoto
    • Organizer
      日台整数論研究集会
    • Place of Presentation
      休暇村気仙沼大島(宮城県・気仙沼市)
    • Year and Date
      2014-09-02
    • Related Report
      2014 Annual Research Report
    • Int'l Joint Research / Invited
  • [Funded Workshop] Iwasawa 20172017

    • Related Report
      2017 Annual Research Report

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Published: 2014-04-04   Modified: 2020-03-30  

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