2011 Fiscal Year Final Research Report
A local and global study of arithmetic varieties determined by arithmetic differential equations
| Project/Area Number |
21654001
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| Research Category |
Grant-in-Aid for Challenging Exploratory Research
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| Allocation Type | Single-year Grants |
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| Research Field |
Algebra
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| Research Institution | Tohoku University |
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Principal Investigator |
TSUZUKI Nobuo 東北大学, 大学院・理学研究科, 教授 (10253048)
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| Co-Investigator(Renkei-kenkyūsha) |
YAMAUCHI Takuya 鹿児島大学, 教育学部, 准教授 (90432707)
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| Project Period (FY) |
2009 – 2011
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| Keywords | 数論幾何、超幾何微分方程式 / カラビ・ヤウ多様体 / 数論的多様体族 / 数論的半安定族 / モノドロミー・重みスペクトル系列 / 保型性 |
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| Research Abstract |
We constructed an arithmetic family of Calabi-Yau varieties on the projective line whose period integral is a generalized hypergeometric function, and determined the relative cohomology of Betti, de Rham, etale and crystalline realizations. This family is a higher dimensional version of Legenedre's family of elliptic curves. The family of Calabi-Yau is obtained by a desingularization which is given explicitly, and has a semistable degeneration at 0. The cohomologies can be calculated by applying weight spectral sequences. Moreover, we have a result on modularity for rational fibers.
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