2001 Fiscal Year Final Research Report Summary
Study on the K3 modular function and its arithmetic aspects
| Project/Area Number |
12640010
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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 Graduate School of Science and Technology at Chiba University, Professor, 大学院・自然科学研究科, 教授 (90009605)
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| Co-Investigator(Kenkyū-buntansha) |
SUGIYAMA Ken-ichi Faculty of Science at Chiba University, Associate Professor, 理学部, 助教授 (90206441)
KITAZUME Masaaki Faculty of Science at Chiba University, Professor, 理学部, 教授 (60204898)
MATSUDA Shigeki Faculty of Science at Chiba University, Assistant, 理学部, 助手 (90272301)
TSUTSUI Toru Faculty of Science at Chiba University, Assistant, 理学部, 助手 (00197732)
ISHIMURA Ryu-ichi Faculty of Science at Chiba University, Professor, 理学部, 教授 (10127970)
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| Project Period (FY) |
2000 – 2001
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| Keywords | Modular function of several variables / K3 surface / period map / Abelian variety / Holonomic system / Hypergeometric Differrential Equation / Bounded symmetric domain / ホロノミックシステム |
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| Research Abstract |
(a) We made a study on the 3 dimensional congruent number problem and rational cuboid problems. We reformed the problem to the study of rational points on a certain type of K3 surfaces and showed the way of construction for some type of rational cuboid.
(b) We studied the quotient of an orthogonal group by the level 2 principal congruence subgroup with respect to a certain typical indefinite quadratic form with 2 positive eigen values. We determined the structure of such quotients.
c We made studies from various aspects on the family of K3 surfaces with a certain fixed structure of the Picard lattice of rank 14. In fact we obtained the following facts :
(i) the explicit model for the member of such a family with a defining equation,
(ii) the explicit moduli space for the periods, that is a bounded symmetric domain of type II,
(iii) description of the differential equation of the periods, that is a holonomic system of rank 8,
(iv) the moduli space can be embedded in the Siegel upper space of degree 8, it is characterized as a certain type of Shimura variety,
(v) the above Shimura variety is induced from the starting Hodge structure of the K3 surface via the construction of Kuga-Satake.
(d) We made an investigation on the Fuchsian differential equation coming from the family of punctured 1 dimensional tori. We obtained several observational results.
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Research Products
(14 results)
