Projective models and automorphism groups on algebraic curves

2021 Fiscal Year Final Research Report

• Project/Area Number: 15K04822
• Research Category: Grant-in-Aid for Scientific Research (C)
• Allocation Type: Multi-year Fund
• Section: 一般
• Research Field: Algebra
• Research Institution: The 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
• Keywords: 自己同型群 / 射影モデル / 代数曲線 / ブリル＝ネーター理論

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.

Free Research Field

代数曲線論

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。