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Projective models and automorphism groups on algebraic curves

Research Project

Project/Area Number 15K04822
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Research Field Algebra
Research InstitutionThe University of Tokushima

Principal Investigator

OHBUCHI Akira  徳島大学, 大学院社会産業理工学研究部(理工学域), 教授 (10211111)

Co-Investigator(Kenkyū-buntansha) 米田 二良  神奈川工科大学, 公私立大学の部局等, 教授 (90162065)
Project Period (FY) 2015-04-01 – 2022-03-31
Project Status Completed (Fiscal Year 2021)
Budget Amount *help
¥4,550,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥1,050,000)
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: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2016: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Fiscal Year 2015: ¥910,000 (Direct Cost: ¥700,000、Indirect Cost: ¥210,000)
Keywords自己同型群 / 射影モデル / 代数曲線 / ブリル=ネーター理論 / 位相幾何学 / 鏡映群 / 計算機 / 自己同型 / フックス群 / reflection group / 位相幾何 / ガロア群
Outline of Final Research Achievements

We calculate automorphism groups of algebraic curves in terms of projective models and finally we have several results on automorphism groups on algebraic curves. For example, we obtain some results on the behavior of the automorphism groups of an analytic family of algebraic curves, on its effects to Weierstrass points, and on Galois points which is a similar concept to Weierstrass points, but is considered as a good criterion for looking the relationship between algebraic curves and Galois theory because we can regard algebraic curves as algebraic function field.
We can calculate several results on these areas. Moreover, we calculate automorphism groups of algebraic curves defined on a positive characteristic field, and in particular, we have severl calculations based on the geometric treatment of modular representations of groups which is mainly by Mitchel, but there are still many questions so unfortunately have to say not so enough.

Academic Significance and Societal Importance of the Research Achievements

代数曲線とその自己同型群は純粋に数学の問題であるが、射影モデルは代数曲線を方程式で定義される対象であると見なす考え方で、方程式が扱われる場、例えば暗号の構成とかディープラーニングなど様々な場との関係が深い。そのため、問題意識は数学に特化した研究であっても、方程式系を扱う事で何らかの応用を求める場合に対する重要な基礎研究と位置付けられるものである。この種の一番有名な応用例はゴレイ符号と言うエラーに対して強い通信の構成理論で、これはMathieu群と言う非常に特別な群の存在により保証されるものである。この様な特殊な群と代数曲線=方程式系の関係を提示するのは社会的にも意義深いと考える。

Report

(8 results)
  • 2021 Annual Research Report   Final Research Report ( PDF )
  • 2020 Research-status Report
  • 2019 Research-status Report
  • 2018 Research-status Report
  • 2017 Research-status Report
  • 2016 Research-status Report
  • 2015 Research-status Report
  • Research Products

    (5 results)

All 2018 2017 2016 2015 2014

All Journal Article (4 results) (of which Int'l Joint Research: 1 results,  Peer Reviewed: 4 results,  Acknowledgement Compliant: 1 results) Presentation (1 results) (of which Invited: 1 results)

  • [Journal Article] Automorphism group of plane curve computed by Galois points, II2018

    • Author(s)
      Harui Takeshi、Miura Kei、Ohbuchi Akira
    • Journal Title

      Proceedings of the Japan Academy, Series A, Mathematical Sciences

      Volume: 94 Issue: 6 Pages: 59-63

    • DOI

      10.3792/pjaa.94.59

    • Related Report
      2018 Research-status Report
    • Peer Reviewed
  • [Journal Article] On γ-hyperelliptic Weierstrass semigroups of genus 6γ+1 and 6γ2017

    • Author(s)
      Jiryo Komeda and Akira Ohbuchi
    • Journal Title

      Bulletin of the Brazilian Mathematical Society

      Volume: 印刷中 Issue: 2 Pages: 209-218

    • DOI

      10.1007/s00574-016-0002-z

    • NAID

      120006802068

    • Related Report
      2017 Research-status Report
    • Peer Reviewed / Acknowledgement Compliant
  • [Journal Article] The Weierstrass Semigroups on Double Cver of genus Two Curves2016

    • Author(s)
      Akira Ohbuchi and Jiryo Komeda
    • Journal Title

      Tukuba Journal of Mathematics

      Volume: vol.38 Pages: 201-206

    • Related Report
      2016 Research-status Report
    • Peer Reviewed / Int'l Joint Research
  • [Journal Article] Automorphism group of planecurve computedby Galois points2014

    • Author(s)
      Ohbuchi,A.
    • Journal Title

      Beitr Algebra Geom

      Volume: in press Issue: 2 Pages: 695-702

    • DOI

      10.1007/s13366-013-0181-3

    • Related Report
      2016 Research-status Report 2015 Research-status Report
    • Peer Reviewed
  • [Presentation] Torres の定理の拡張について2015

    • Author(s)
      大渕 朗
    • Organizer
      ワークショップ「ホッジ理論と代数幾何学」
    • Place of Presentation
      東京電機大学・東京千住キャンパス2号館5階2505教室(東京都足立区)
    • Year and Date
      2015-08-07
    • Related Report
      2015 Research-status Report
    • Invited

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Published: 2015-04-16   Modified: 2023-01-30  

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