2018 Fiscal Year Final Research Report
A experimental study of arithmetic local systems with geometric origins and unsolved problems in arithmetic geometry
| Project/Area Number |
15K13422
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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, 理学研究科, 教授 (10253048)
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| Research Collaborator |
Yamauchi Takuya 東北大学, 大学院理学研究科
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| Project Period (FY) |
2015-04-01 – 2019-03-31
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| Keywords | 数論的一般超幾何カラビ・ヤウ多様体族 / 一般超幾何関数 / 剛性Calabi-Yau多様体 / 保型性 / 2進モデル / F-アイソクリスタル / Newton多角形 / 代数曲線族のisotriviality |
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| Outline of Final Research Achievements |
While there are lots of unsolved problems in arithmetic geometry, it is difficult to construct explicit examples because of abstraction. Using the arithmetic family of higher dimensional Calabi-Yau varieties, which has been constructed by the author, over the projective line of invertible 2, (1) we show an irreducible component of the degenerated fiber is a rigid Calabi-Yau variety over the field of rational numbers, and prove the modularity and the algebraicity of cohomology classes, and (2) study a 2-adic model and construct a K3 surface over a real quadratic field with everywhere good reduction. We also study p-adic properties of Frobenius actions arithmetically, and prove the constancy of Newton polygons of arbitrary F-isocrystals on Abelian varieties. This result has not been known so far.
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| Free Research Field |
数論幾何学
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| Academic Significance and Societal Importance of the Research Achievements |
数論幾何学の未解決問題は極めて抽象的であり、意味のある具体的な例を構成することでさえ困難を極める。これまでの研究で構成した数論的な高次元の一般超幾何Calabi-Yau多様体族を用いて、保型性や代数的サイクルの様子を具体的に考察し、特別な場合に未解決予想を検証することができた。また、Frobenius作用のp進的な性質を発見して、アーベル多様体という数論幾何学における最も基本的な対象に対して新たな現象を発見することができ、幾何学的な応用を与えた。このことにより、数論幾何学に新たな視点を提出することができた。
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