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Studies on the internal structure of algebraic function fields - from the viewpoint of Galois point studies

Research Project

Project/Area Number 19K03441
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeMulti-year Fund
Section一般
Review Section Basic Section 11010:Algebra-related
Research InstitutionNiigata University

Principal Investigator

Takahashi Takeshi  新潟大学, 自然科学系, 准教授 (60390431)

Project Period (FY) 2019-04-01 – 2024-03-31
Project Status Completed (Fiscal Year 2023)
Budget Amount *help
¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000)
Fiscal Year 2021: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2020: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2019: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Keywordsガロア点 / 準ガロア点 / ガロア直線 / 同時ガロア点 / 代数関数体の内部構造 / 射影代数多様体の自己同型群 / 代数関数体 / 自己同型群 / 射影的代数曲線 / 弱ガロア・ワイエルシュトラス点 / 代数曲線 / ガロア・ワイエルシュトラス点
Outline of Research at the Start

代数関数体にどのような部分体が含まれているのかを, 代数幾何を用いて調べたい. 平面代数曲線の関数体を調べるためにガロア点が研究されてきた. より広いクラスの代数関数体をより詳しく調べるためには, ガロア点の概念をどのように一般化すべきだろうか.
平面代数曲線に対する準ガロア点, 代数曲線の弱ガロア・ワイエルシュトラス点, およびそれらの高次元化の研究を行う. 具体的には, それらの個数と分布, 付随する群を調べる. さらに, 得られた結果を射影多様体の分類や自己同型群の研究などに応用する. ガロア点理論の研究者を訪問して共同研究を行う. また, ワークショップを開催して意見交換を行う.

Outline of Final Research Achievements

The study related to "Galois points for algebraic plane curves", introduced by Hisao Yoshihara (Niigata Univ.) have been investigated steadily, with several generalizations.
S. Fukazawa (Yamagata Univ.), K. Miura (Ube National College of Technology) and I defined "quasi-Galois points" for plane curves and studied the groups associated with quasi-Galois points, the number and distribution of quasi-Galois points. J. Komeda (Kanagawa Institute of Technology), S. Kato (Niigata Univ.) and I studied the number and distribution of "Galois lines" for nonsingular curves of degree 6 in 3-dimensional projective space, and determined the maximum numbers and distribution of such lines. A. Ikeda (Niigata Univ.) and I defined "simultaneous Galois points" for reducible and reduced algebraic plane curves. The number of simultaneous Galois points could be determined when each irreducible component of an algebraic curve is a non-singular plane curve not an elliptic curve.

Academic Significance and Societal Importance of the Research Achievements

射影平面内の代数曲線についての準ガロア点の個数と分布について多くの成果をあげることができ、結果を整理した論文を投稿することができた。3次元射影空間内の射影代数曲線に対するガロア直線について、その本数や分布を調べる方法を開発した。また、既約平面曲線に限らず、可約な平面曲線に対して、そのガロア点である同時ガロア点を定義し、それを研究する基礎を与えた。つまり、本研究課題において、ガロア点に関係する新しい概念をいくつか提供し、それらの基本定理を示すことができた。

Report

(6 results)
  • 2023 Annual Research Report   Final Research Report ( PDF )
  • 2022 Research-status Report
  • 2021 Research-status Report
  • 2020 Research-status Report
  • 2019 Research-status Report
  • Research Products

    (9 results)

All 2024 2023 2022 2019

All Journal Article (5 results) (of which Peer Reviewed: 5 results) Presentation (4 results) (of which Invited: 1 results)

  • [Journal Article] Simultaneous Galois points for a reducible plane curve consisting of nonsingular components2024

    • Author(s)
      Aki Ikeda and Takeshi Takahashi
    • Journal Title

      Kodai Mathematical Journal

      Volume: -

    • Related Report
      2023 Annual Research Report
    • Peer Reviewed
  • [Journal Article] Galois lines for a canonical curve of genus 4, I: Non-skew cyclic lines2022

    • Author(s)
      Jiryo Komeda and Takeshi Takahashi
    • Journal Title

      Rendiconti del Seminario Matematico della Universita di Padova

      Volume: -

    • Related Report
      2022 Research-status Report
    • Peer Reviewed
  • [Journal Article] Galois lines for a canonical curve of genus 4, II: skew cyclic lines2022

    • Author(s)
      Jiryo Komeda and Takeshi Takahashi
    • Journal Title

      Rendiconti del Seminario Matematico della Universita di Padova

      Volume: -

    • Related Report
      2022 Research-status Report
    • Peer Reviewed
  • [Journal Article] Algebraic curves admitting the same Galois closure for two projections2022

    • Author(s)
      Satoru Fukasawa, Kazuki Higashine and Takeshi Takahashi
    • Journal Title

      Annali di Matematica Pura ed Applicata (1923-)

      Volume: - Issue: 5 Pages: 2055-2061

    • DOI

      10.1007/s10231-022-01191-0

    • Related Report
      2021 Research-status Report
    • Peer Reviewed
  • [Journal Article] Number of weak Galois-Weierstrass points with Weierstrass semigroups generated by two elements2019

    • Author(s)
      Komeda, Jiryo and Takahashi, Takeshi
    • Journal Title

      Journal of the Korean Mathematical Society

      Volume: 56(6) Pages: 1463-1474

    • Related Report
      2019 Research-status Report
    • Peer Reviewed
  • [Presentation] 可約平面曲線の同時ガロア点2023

    • Author(s)
      高橋剛
    • Organizer
      第21回代数曲線論シンポジウム
    • Related Report
      2023 Annual Research Report
  • [Presentation] 種数4の標準曲線に対する skew cyclic line の本数2022

    • Author(s)
      高橋剛
    • Organizer
      新潟代数セミナー
    • Related Report
      2022 Research-status Report
  • [Presentation] Galois lines for a space curve of genus 42019

    • Author(s)
      高橋剛
    • Organizer
      第17回代数曲線論シンポジウム 於神奈川工科大学アクティブラーニング横浜
    • Related Report
      2019 Research-status Report
    • Invited
  • [Presentation] 種数4曲線のガロアラインについて, その22019

    • Author(s)
      高橋剛
    • Organizer
      Workshop on Galois point and related topics, 山形大学理学部
    • Related Report
      2019 Research-status Report

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Published: 2019-04-18   Modified: 2025-01-30  

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