Integral structures of arithmetic differential equations and geometries behind them
| Project/Area Number |
24654002
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| Research Category |
Grant-in-Aid for Challenging Exploratory Research
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| Allocation Type | Multi-year Fund |
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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, 理学(系)研究科(研究院), 教授 (10253048)
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| Co-Investigator(Kenkyū-buntansha) |
YAMAUCHI Takuya 鹿児島大学, 教育学部, 准教授 (90432707)
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| Co-Investigator(Renkei-kenkyūsha) |
TAKAHASHI Nobuyoshi 広島大学, 大学院理学研究科, 准教授 (60301298)
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| Project Period (FY) |
2012-04-01 – 2015-03-31
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| Project Status |
Completed (Fiscal Year 2014)
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| Budget Amount *help |
¥3,900,000 (Direct Cost: ¥3,000,000、Indirect Cost: ¥900,000)
Fiscal Year 2014: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2013: ¥1,170,000 (Direct Cost: ¥900,000、Indirect Cost: ¥270,000)
Fiscal Year 2012: ¥1,560,000 (Direct Cost: ¥1,200,000、Indirect Cost: ¥360,000)
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| Keywords | 超幾何関数 / 剛性カラビ・ヤウ多様体 / 半安定的退化 / 整係数コホモロジー / ガロア表現 / 保型性 / 2進還元 / 数論幾何学 / カラビ・ヤウ多様体族 / 退化族 / 国際研究者交流(イタリア) / 数論幾何 / 超幾何微分方程式 / カラビ・ヤウ多様体 / 整構造 / 数論幾 / 超幾何微分方程 / 局所系の整構造 |
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| Outline of Final Research Achievements |
We studied properties of the arithemtic family of Calabi-Yau varieties, constructed by the representative of this research, for which the period integral is a generalized hypergeometric functions. In particular, if the dimension is odd, we found a semistable family around a degenerated fiber such that the number of irreducible components of the special fiber is two among which the one is rational and the other has an interesting natures with respect to arithmetic geometry. In particuler, we proved the modularity of the special fiber in dimension 3.
In the case of dimension 2, we constructed a semistable family over an extension
of Z which is ramified only at 2 and got a K3 surface over an algebraic number field such that it has a good reduction everywhere.
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Report
(4 results)
Research Products
(27 results)
