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Interpretation of geometric objects by minimal complexes and arithmetic applications

Research Project

Project/Area Number 16K17572
Research Category

Grant-in-Aid for Young Scientists (B)

Allocation TypeMulti-year Fund
Research Field Algebra
Research InstitutionKyoto University

Principal Investigator

Ishitsuka Yasuhiro  京都大学, 理学研究科, 特定助教 (50761136)

Project Period (FY) 2016-04-01 – 2020-03-31
Project Status Completed (Fiscal Year 2019)
Budget Amount *help
¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000)
Fiscal Year 2019: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2018: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2017: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Fiscal Year 2016: ¥1,040,000 (Direct Cost: ¥800,000、Indirect Cost: ¥240,000)
Keywords数論的不変式論 / 行列式表示 / 平面曲線 / 局所大域性 / 軌道指数和 / ヤコビ多様体 / フェルマー曲線 / 指数和 / 余正則空間 / 空間六次曲線 / 局所大域原理 / 平面四次曲線 / 変曲点 / 平面三次曲線 / 指数和の計算 / 法ガロア表現 / Fermatの四次曲線 / Kleinの四次曲線 / 概均質ベクトル空間 / 線形行列式表示 / 数論幾何学 / 環論 / 代数群の線形表現 / 極小射影分解
Outline of Final Research Achievements

With respect to correspondence between arithmetic objects and orbits of linear representations, we interpreted rotationally symmetric determinantal representations of smooth plane cubics, and deduced the average behavior. We obtained results related to geometric objects in plane or space curves from arithmetic viewpoint. For example, we determined the Galois action on 4-torsion points of Jacobian varieties of Fermat quartic curves; we constructed a plane quartic failing the local-global property for the existence of bitangents; we proved the local-global property for the existence of flexes in smooth plane cubics in a strong sense. We also determined the orbital exponential sum of the space of binary quartic forms.

Academic Significance and Societal Importance of the Research Achievements

新規に構成した対応と以前の類似の対応を比較すると、数論的不変式論の手法では、Jacobi多様体の有理点とPicard群の差異のような、微妙かつ重要な差を表すことができることを示唆している。また周辺の問題を研究することで、古典的な代数幾何に現れる対象のもつ数論的な側面が明らかになったほか、それらの研究での数式処理システムの重要性を再確認できる結果になった。軌道指数和についても、BhargavaとHoの種数1の曲線と軌道との対応を利用することが必要になったほか、古典的に見いだされていた不変式の解釈も予期せず用いる形になり、数論的不変式論において古典論の重要性を再確認できるものとなった。

Report

(5 results)
  • 2019 Annual Research Report   Final Research Report ( PDF )
  • 2018 Research-status Report
  • 2017 Research-status Report
  • 2016 Research-status Report
  • Research Products

    (22 results)

All 2019 2018 2017 2016 Other

All Int'l Joint Research (6 results) Journal Article (5 results) (of which Peer Reviewed: 5 results,  Open Access: 2 results) Presentation (10 results) (of which Int'l Joint Research: 1 results,  Invited: 4 results) Funded Workshop (1 results)

  • [Int'l Joint Research] トロント大学(カナダ)

    • Related Report
      2019 Annual Research Report
  • [Int'l Joint Research] サウスカロライナ大学(米国)

    • Related Report
      2019 Annual Research Report
  • [Int'l Joint Research] トロント大学(カナダ)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] サウスカロライナ大学(米国)

    • Related Report
      2018 Research-status Report
  • [Int'l Joint Research] University of South Carolina(米国)

    • Related Report
      2017 Research-status Report
  • [Int'l Joint Research] University of Oxford(英国)

    • Related Report
      2017 Research-status Report
  • [Journal Article] Explicit calculation of the mod 4 Galois representation associated with the Fermat quartic2019

    • Author(s)
      Ishitsuka Yasuhiro、Ito Tetsushi、Ohshita Tatsuya
    • Journal Title

      International Journal of Number Theory

      Volume: - Issue: 04 Pages: 1-25

    • DOI

      10.1142/s1793042120500451

    • Related Report
      2019 Annual Research Report
    • Peer Reviewed
  • [Journal Article] On algorithms to obtain linear determinantal representations of smooth plane curves of higher degree2019

    • Author(s)
      Ishitsuka Yasuhiro、Ito Tetsushi、Ohshita Tatsuya
    • Journal Title

      JSIAM Letters

      Volume: 11 Issue: 0 Pages: 9-12

    • DOI

      10.14495/jsiaml.11.9

    • NAID

      130007606280

    • ISSN
      1883-0609, 1883-0617
    • Related Report
      2018 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] The local-global principle for symmetric determinantal representations of smooth plane curves in characteristic two2017

    • Author(s)
      Yasuhiro Ishitsuka and Tetsushi Ito
    • Journal Title

      Journal of Pure and Applied Algebra

      Volume: 221 Issue: 6 Pages: 1316-1321

    • DOI

      10.1016/j.jpaa.2016.09.013

    • Related Report
      2016 Research-status Report
    • Peer Reviewed
  • [Journal Article] An algorithm to obtain linear determinantal representations of smooth plane cubics over finite fields2016

    • Author(s)
      Yasuhiro Ishitsuka
    • Journal Title

      JSIAM Letters

      Volume: 8 Issue: 0 Pages: 69-72

    • DOI

      10.14495/jsiaml.8.69

    • NAID

      130005285603

    • ISSN
      1883-0609, 1883-0617
    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Open Access
  • [Journal Article] On the symmetric determinantal representations of the Fermat curves of prime degree2016

    • Author(s)
      Yasuhiro Ishitsuka and Tetsushi Ito
    • Journal Title

      International Journal of Number Theory (IJNT)

      Volume: 12 Issue: 04 Pages: 955-967

    • DOI

      10.1142/s1793042116500597

    • Related Report
      2016 Research-status Report
    • Peer Reviewed
  • [Presentation] 平面4次曲線の双接線に関する局所大域性2019

    • Author(s)
      石塚裕大
    • Organizer
      日本応用数理学会2019年研究部会連合発表会 「数論アルゴリズムとその応用」(JANT)セッション
    • Related Report
      2018 Research-status Report
  • [Presentation] 平面三次曲線の変曲点の局所大域性について2018

    • Author(s)
      石塚裕大
    • Organizer
      慶應代数セミナー
    • Related Report
      2018 Research-status Report
    • Invited
  • [Presentation] 平面三次曲線の変曲点の局所大域性について2018

    • Author(s)
      石塚裕大
    • Organizer
      第228回 北陸数論セミナー
    • Related Report
      2018 Research-status Report
  • [Presentation] 平面4次曲線の線形行列式表示を計算するアルゴリズムについて2018

    • Author(s)
      大下達也
    • Organizer
      日本応用数理学会2018年研究部会連合発表会 「数論アルゴリズムとその応用」(JANT)セッション
    • Related Report
      2017 Research-status Report
  • [Presentation] Computation on linear determinantal representations2017

    • Author(s)
      石塚裕大
    • Organizer
      神戸整数論集会2017
    • Related Report
      2017 Research-status Report
    • Int'l Joint Research / Invited
  • [Presentation] Arithmetic of rotationally symmetric determinantal representations2017

    • Author(s)
      Yasuhiro Ishitsuka
    • Organizer
      Arithmetic Geometry and Related Topics
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] 三次曲線の回転対称な行列式表示について2017

    • Author(s)
      石塚裕大
    • Organizer
      第16回北陸数論研究集会
    • Related Report
      2017 Research-status Report
    • Invited
  • [Presentation] 大域体での線形行列式表示2016

    • Author(s)
      石塚裕大
    • Organizer
      九州代数セミナー
    • Place of Presentation
      九州大学
    • Year and Date
      2016-12-09
    • Related Report
      2016 Research-status Report
  • [Presentation] 平面曲線の行列式表示の数論的側面2016

    • Author(s)
      石塚裕大
    • Organizer
      第10回福岡数論研究集会
    • Place of Presentation
      九州大学
    • Year and Date
      2016-08-09
    • Related Report
      2016 Research-status Report
  • [Presentation] Arithmetic of linear determinantal representations; I, II2016

    • Author(s)
      石塚裕大
    • Organizer
      第18回静岡代数学セミナー
    • Place of Presentation
      静岡大学
    • Year and Date
      2016-07-08
    • Related Report
      2016 Research-status Report
  • [Funded Workshop] 神戸整数論集会2017(Kobe Number Theory Workshop 2017)2017

    • Related Report
      2017 Research-status Report

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Published: 2016-04-21   Modified: 2022-02-22  

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