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Infinite product presentation of the Mumford form and special values of geometric zeta functions

Research Project

Project/Area Number 26400018
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Algebra
Research InstitutionSaga University

Principal Investigator

ICHIKAWA Takashi  佐賀大学, 工学(系)研究科(研究院), 教授 (20201923)

Co-Investigator(Kenkyū-buntansha) 中川 泰宏  佐賀大学, 工学(系)研究科(研究院), 教授 (90250662)
庄田 敏宏  佐賀大学, 教育学部, 准教授 (10432957)
Project Period (FY) 2014-04-01 – 2017-03-31
Project Status Completed (Fiscal Year 2016)
Budget Amount *help
¥4,810,000 (Direct Cost: ¥3,700,000、Indirect Cost: ¥1,110,000)
Fiscal Year 2016: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2015: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2014: ¥1,690,000 (Direct Cost: ¥1,300,000、Indirect Cost: ¥390,000)
Keywords代数曲線 / モジュライ空間 / Chern-Simons不変量 / Arakelov理論 / Deligne-Riemann-Roch同型写像 / Schottky群 / Ruelleゼータ関数 / 数論幾何 / Riemann-Roch同型 / ゼータ関数 / アラケロフ幾何 / チャーン・サイモンズ不変量 / リウヴィル場 / 算術的リーマン・ロッホの定理 / アインシュタイン・ケーラー計量 / 極小曲面 / リーマン面 / ショットキー一意化 / リーマン・ロッホの定理 / 双曲3次元多様体 / チャーン・サイモンズ理論
Outline of Final Research Achievements

By the arithmetic Schottky-Mumford uniformization theory, we proved the arithmeticity of Chern-Simons invariants. Using this result together with the Arakelov theory in arithmetic geometry and the theory of Zograf, Mcintyre-Takhatajan on the classical Liouville field theory, we gave an infinite product presentation of the Deligne-Riemann-Roch isomorphism which expresses the Chern-Simons line bundle. As its application, we express the special values of the Ruelle zeta functions of Schottky groups as the products of period integrals and discriminants. This result gives an analog of the Deligne conjecture on
such geometric zeta values.

Report

(4 results)
  • 2016 Annual Research Report   Final Research Report ( PDF )
  • 2015 Research-status Report
  • 2014 Research-status Report
  • Research Products

    (17 results)

All 2017 2016 2015 2014

All Journal Article (6 results) (of which Peer Reviewed: 6 results,  Acknowledgement Compliant: 3 results,  Open Access: 2 results) Presentation (11 results) (of which Int'l Joint Research: 1 results,  Invited: 10 results)

  • [Journal Article] On hyperelliptic minimal surfaces with even genus2016

    • Author(s)
      N. Ejiri, T. Shoda
    • Journal Title

      Proceedings of the 4th international colloquium on Differential Geometry and its related fields

      Volume: なし Pages: 129-138

    • Related Report
      2016 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Minimal surfaces with two ends which have the least total absolute curvature2016

    • Author(s)
      Shoichi Fujimori and Toshihiro Shoda
    • Journal Title

      Pacific Journal of Mathematics

      Volume: 282 Issue: 1 Pages: 107-144

    • DOI

      10.2140/pjm.2016.282.107

    • Related Report
      2016 Annual Research Report 2015 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] On the examples of Nill and Paffenholz2015

    • Author(s)
      Yasuhiro Nakagawa
    • Journal Title

      International Journal of Mathematics

      Volume: 26 Issue: 04 Pages: 673687-673687

    • DOI

      10.1142/s0129167x15400078

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Open Access / Acknowledgement Compliant
  • [Journal Article] A remark on limits of triply periodic minimal surfaces of genus 32015

    • Author(s)
      N. Ejiri, S. Fujimori and T. Shoda
    • Journal Title

      Topology and its Applications

      Volume: 196 Pages: 880-903

    • DOI

      10.1016/j.topol.2015.05.014

    • Related Report
      2015 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] Algebraic and rigid geometry on the Schottky problem2014

    • Author(s)
      Takashi Ichikawa
    • Journal Title

      J. reine angew. Math.

      Volume: 印刷中 Issue: 705 Pages: 23-33

    • DOI

      10.1515/crelle-2013-0059

    • Related Report
      2015 Research-status Report
    • Peer Reviewed
  • [Journal Article] Vector-valued p-adic Siegel modular forms2014

    • Author(s)
      Takashi ICHIKAWA
    • Journal Title

      J. reine angew. Math

      Volume: (印刷中) Issue: 690 Pages: 35-49

    • DOI

      10.1515/crelle-2012-0066

    • Related Report
      2014 Research-status Report
    • Peer Reviewed / Open Access
  • [Presentation] Arithmetic invariants for families of algebraic curves2017

    • Author(s)
      市川尚志
    • Organizer
      2次微分の幾何とその周辺
    • Place of Presentation
      東京大学
    • Year and Date
      2017-01-11
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] 概正則ジーゲル保型形式の代数的理論と概超収束ジーゲル保型形式2016

    • Author(s)
      市川尚志
    • Organizer
      p進保型形式、p進ガロア表現と関連する話題
    • Place of Presentation
      佐賀大学
    • Year and Date
      2016-10-17
    • Related Report
      2016 Annual Research Report
  • [Presentation] Polylogarithm functions on algebraic curves2016

    • Author(s)
      市川尚志
    • Organizer
      2016大分整数論研究集会
    • Place of Presentation
      ホルトホール大分
    • Year and Date
      2016-10-09
    • Related Report
      2016 Annual Research Report
    • Invited
  • [Presentation] 極小曲面の存在と一意性について2016

    • Author(s)
      庄田 敏宏
    • Organizer
      Aspects of Differential Geometry
    • Place of Presentation
      東北大学
    • Year and Date
      2016-03-09
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] ある種のトーリック束の上のアインシュタイン・ケーラー計量の存在問題について2016

    • Author(s)
      中川泰宏
    • Organizer
      2016名城幾何学研究集会「幾何構造の深化」
    • Place of Presentation
      名城大学
    • Year and Date
      2016-03-01
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] 3次元Euclid空間内の全曲率有限な完備極小曲面の存在と一意性について2016

    • Author(s)
      庄田 敏宏
    • Organizer
      2016名城大学幾何学研究集会「種々の幾何構造の深化」
    • Place of Presentation
      名城大学
    • Year and Date
      2016-03-01
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] Algebraic theory of nearly holomorphic modular forms2016

    • Author(s)
      Takashi Ichikawa
    • Organizer
      RIMS研究集会「Automorphic Forms, Automorphic L-functions and Related Topics」
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2016-02-02
    • Related Report
      2015 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Riemann-Roch isomorphisms on the moduli spaces of curves2015

    • Author(s)
      Takashi Ichikawa
    • Organizer
      Moduli spaces of abelian varieties and curves, and related analysis
    • Place of Presentation
      東京大学
    • Year and Date
      2015-12-15
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] 三重周期極小曲面の周期性の崩壊とラメラ構造について2015

    • Author(s)
      庄田 敏宏
    • Organizer
      部分多様体幾何とリー群作用2015
    • Place of Presentation
      東京理科大学
    • Year and Date
      2015-09-07
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] 非対称アインシュタイン・ケーラートーリックFano多様体について2015

    • Author(s)
      中川泰宏
    • Organizer
      RIMS研究集会「幾何学・組合せ論に現れる環と代数構造」
    • Place of Presentation
      京都大学数理解析研究所
    • Year and Date
      2015-06-09
    • Related Report
      2015 Research-status Report
    • Invited
  • [Presentation] Riemann-Roch isomorphisms as infinite products2014

    • Author(s)
      Takashi Ichikawa
    • Organizer
      Bundles over surfaces and Eisenstein periods for loop groups
    • Place of Presentation
      九州大学
    • Year and Date
      2014-07-01
    • Related Report
      2014 Research-status Report
    • Invited

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Published: 2014-04-04   Modified: 2018-03-22  

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