Arithmetic aspects of Calabi-Yau surfaces and the hypergeometric system

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In this project we aimed to find a new aspect of the period map for the K3 surfaces based on the hypergeometric differential equations. We expected arithmetic applications of our approach also. For it, it is basic to obtain explicit representations of the modular functions those arise as the inverse of the period map. There are various foregoing studies for the moduli of abelian surfaces. But there was no exact result to describe the modular maps based on the geometric back ground. We found a good description of theparameter space for it by considering a family of elliptic K3 surfaces those are Hodge equivalent to the family of abelian surfaces. By this idea we succeeded to solve the above problem. The main result will be published as"Modular maps for the family of abelian surfaces via K3 surfaces", Math. Nachrichten (2014)(in printing, joint work with A. Nagano).