Study on the K3 modular function and its arithmetic aspects

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(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.