2013 Fiscal Year Final Research Report
Motivic structure of nilpotent completions of modular groups
| Project/Area Number |
23540021
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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 |
ICHIKAWA Takashi 佐賀大学, 工学(系)研究科(研究院), 教授 (20201923)
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| Co-Investigator(Kenkyū-buntansha) |
UEHARA Tsuyoshi 佐賀大学, 大学院・工学系研究科, 教授 (80093970)
MIYAZAKI Chikashi 佐賀大学, 大学院・工学系研究科, 教授 (90229831)
TERAI Naoki 佐賀大学, 文化教育学部, 教授 (90259862)
HIROSE Susumu 東京理科大学, 理工学部, 准教授 (10264144)
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| Project Period (FY) |
2011 – 2013
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| Keywords | 数論幾何 / 代数曲線 / アーベル多様体 / モジュライ空間 / ショットキー問題 / モジュラー形式 / ゼータ関数 |
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| Research Abstract |
By studying arithmetic geometry of algebraic curves, abelian varieties and their moduli spaces, we obtained the following results. 1. We constructed a theory of Hecke operators on elliptic modular motives, and as its application, we showed the algebraicity of multiple modular L-values. 2. Using rigid analysis, we gave a solution to the Schottky problem, namely a condition that abelian varieties become Jacobi varieties. 3. We constructed a basic theory of p-adic vector-valued Siegel modular forms. Further, we gave p-adic versions of Shimura's nearly holomorphic vector-valued Siegel modular forms and showed the algebraicity of their values at CM points. 4. By using the arithmetic Schottky uniformization theory, we showed the arithmeticity of the special values for geometric zeta functions of hyperbolic 3-manifolds.
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