Arithmetic research of hypergeometric differential equation and its Schwarz map
| Project/Area Number |
17540011
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Chiba University |
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Principal Investigator |
SHIGA Hironori Chiba University, Graduate school of Science, Professor (90009605)
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| Co-Investigator(Kenkyū-buntansha) |
KITAZUME Masaaki Chiba University, Graduate school of Science, Professor (60204898)
SUGIYAMA Kenichi Chiba University, Graduate school of Science, Associate Professor (90206441)
MATSUDA Shigeki Chiba University, Graduate school of Science, Associate Professor (90272301)
安田 正実 千葉大学, 理学部, 教授 (00041244)
多田 充 千葉大学, メディア基盤センター, 助教授 (20303331)
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| Project Period (FY) |
2005 – 2007
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| Project Status |
Completed (Fiscal Year 2007)
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| Budget Amount *help |
¥3,800,000 (Direct Cost: ¥3,500,000、Indirect Cost: ¥300,000)
Fiscal Year 2007: ¥1,300,000 (Direct Cost: ¥1,000,000、Indirect Cost: ¥300,000)
Fiscal Year 2006: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2005: ¥1,500,000 (Direct Cost: ¥1,500,000)
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| Keywords | Arithmetic Geometric Mean / Schwarz map / Period Map / Picard Modular form / Hypergeometric Function / Abelian variety / Complex multiplication / アーベル多様体の虚数乗法 / 超幾何微分方程式 / 保型形式 / 周期 / テータ函数 / 超幾何函数 |
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| Research Abstract |
1) In 1991 Jonathan and Peter Borweins have discovered a nice variant of the GaussAGM (Arithmetic Geometric Mean). It has been an almost unique nice result about GauasAGM. There is no successful study about AGM functions in several variables. In our project we have succeeded to established an AGM function of two variables that is an extension of Borweins' result See the article on J. N. T. 2007.
2) We made an investigation on the values of the Schwarz map of the Gauss hypergeometric differential eqations. It can be regarded as a ratio of periods of corresponding hypergeometric curves of the differentioal of the second kind. In our study we fixed the family of hypergeometric curves. And we considered the albebraic values of the Schwarz maps At algebraic points for various differentials of the second kind at a same time. We obtained a general results and some illustrating Examples. See the article Birkhauser Monograph 2007.
3) Together with graduate students of the head investigator we made a study of the Schwarz map for the hypergeometric Differential equations of several variables. We obtained some analogous results as 2). The article is under preparation. And We made a investigation about the period map of families of algebraic K3 surfaces. Especially a family with some specific Transcendental lattice of rank 4. That is parametrized by a Hilbert modular surface for the square root of 2. This study is now in Preparation also.
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Report
(4 results)
Research Products
(22 results)
