Study on the K3 modular function and its arithmetic aspects

K3模函数及其算术问题的研究

基本信息

批准号：
12640010
负责人：
SHIGA Hironori
金额：
$ 2.18万
依托单位：
CHIBA UNIVERSITY
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2001
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640010/
关键词：
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ミラ対称性超幾何微分方程式モノドロミー格子志村多様体

项目摘要

(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.

(a)我们对三维全等数问题和有理长方体问题进行了研究。我们将问题改造为某类K3曲面上有理点的研究，并给出了某类有理长方体的构造方法。(b)我们研究了由2级主同余子群对正交群的商到具有 2 个正特征值的某个典型的不定二次形式。我们确定了这些商的结构。c 我们对具有一定固定结构的 14 阶 Picard 格子的 K3 曲面族进行了多方面的研究。实际上我们得到了以下事实：(i)具有定义方程的此类族的成员，(ii) 周期的显式模空间，即 II 型有界对称域，(iii) 周期微分方程的描述，即完整的秩系统8,(iv)模空间可以嵌入到8次Siegel上空间中，它被表征为某种类型的Shimura簇，(v)上述Shimura簇是从K3表面的起始Hodge结构推导出来的Kuga-Satake的构造。(d)我们对来自穿孔一维环面族的Fuchsian微分方程进行了研究。我们获得了一些观察结果。