2021 Fiscal Year Final Research Report
Arithmetic geometry of the moduli spaces of algebraic curves and abelian varieties, and its applications
| Project/Area Number |
17K05179
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Multi-year Fund |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Saga University |
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Principal Investigator |
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| Co-Investigator(Kenkyū-buntansha) |
庄田 敏宏 関西大学, システム理工学部, 教授 (10432957)
中村 健太郎 佐賀大学, 理工学部, 准教授 (90595993)
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| Project Period (FY) |
2017-04-01 – 2022-03-31
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| Keywords | 代数曲線 / アーベル多様体 / モジュライ空間 / リーマン・ロッホ同型写像 / マンフォード曲線 / アーベル微分 / 周期積分 / KP階層 |
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| Outline of Final Research Achievements |
Using an explicit formula of abelian differentials on generalized Tate curves, we give infinite product expressions of the Riemann-Roch isomorphism and the Mumford isomorphism defined for families of algebraic curves, and study the degeneration of quasi-periodic solutions of the KP hierarchy with application to its solutions obtained as mixtures of quasi-periodic and soliton solutions. Furthermore, we construct the universal Mumford curve, and show explicit formulas of its (universal) abelian differentials, period integrals and compactified Jacobians. As a nonabelian extension of this result, we give asypmtotic explicit formulas of unipotent peirods using multiple polylogarithm functions and zeta values.
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| Free Research Field |
代数曲線とそのモジュライ空間に関する数論幾何的研究と応用
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| Academic Significance and Societal Importance of the Research Achievements |
代数曲線、アーベル多様体のモジュライ空間の数論幾何的研究に新しい知見を与えると共に、弦理論の測度関数、ソリトン方程式、ファインマン積分などの理論物理や数理物理への応用も視野に入れることができた。
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