A study on automorphic forms of several variables with symmetries of level structure
| Project/Area Number |
17K05186
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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 | Kyoto Sangyo University |
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Principal Investigator |
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| Project Period (FY) |
2017-04-01 – 2021-03-31
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| Project Status |
Completed (Fiscal Year 2020)
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| Budget Amount *help |
¥4,420,000 (Direct Cost: ¥3,400,000、Indirect Cost: ¥1,020,000)
Fiscal Year 2019: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
Fiscal Year 2018: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
Fiscal Year 2017: ¥1,430,000 (Direct Cost: ¥1,100,000、Indirect Cost: ¥330,000)
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| Keywords | Borcherds積 / 対称性 / 階層構造 / ヤコビ保型形式 / ジーゲル保型形式 / モジュラー曲線 / 完全再生可能関数 / ヤコビ形式 / ベクトル系 / 無限積 / 保型形式 / 直交群 / モジュラー多項式 / アフィン・リー環 / テータリフト / 2次ジーゲル保型形式 / 斎藤-黒川リフティング / 多変数保型形式 / 四元数ユニタリ群 |
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| Outline of Final Research Achievements |
We investigated on relations between certain symmetries of automorphic forms of several variables and infinite product expansions of Borcherds type. Our goal is to show that a family of automorphic forms with multiplicative symmetries of level structure has an infinite product expansion of Borcherds type. In this study, we showed that a family of Jacobi forms with multiplicative symmetries of level structure has an infinite product expansion. By using this result, we also showed that a family of Siegel modular forms of degree 2 with multiplicative symmetries of level structure has an infinite product expansion. We also proved that a plane curve with a single symmetry of Hecke type is in fact a modular curve, and that completely replicable functions are characterized by multiplicative symmetries of level structure.
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| Academic Significance and Societal Importance of the Research Achievements |
多変数保型形式の整数論は近年研究が進んでいるものの、いまだ未解決の問題が多く、研究テーマとして大変興味深い分野である。多変数保型形式の中でも、Borcherds型の無限積展開を持つものは、代数幾何学や数理物理学とも関係して、重要な研究対象である。本研究では、Borcherds型の無限積展開を持つことと、本研究で新しく導入された階層構造を持つHecke型の乗法対称性が同値であることを、ヤコビ保型形式と２次ジーゲル保型形式の場合に証明した。与えられた保型形式がBorcherds型の無限積展開を持つかどうかは、判定が難しい問題だったが、本研究によって、判定する新しい方法が得られた。
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Report
(5 results)
Research Products
(10 results)
