Arithmetic aspects of Calabi-Yau surfaces and the hypergeometric system

在这个项目中，我们的目标是找到一个新的方面的周期映射的K3曲面的超几何微分方程的基础上。我们也期望我们的方法的算术应用。对于它，它是基本的，以获得明确表示的模函数，这些出现作为逆的周期映射。对于交换曲面的模，已有各种各样的研究。但目前还没有确切的结果来描述基于几何背景的模块化地图。我们通过考虑一类与交换曲面族Hodge等价的椭圆K_3曲面族，找到了它的参数空间的一个很好的描述。通过这个想法我们成功地解决了上述问题。主要结果将发表为“Modular maps for the family of abelian surfaces via K3 surfaces”，Math.Nachrichten（2014）（印刷中，与A. Nagano）。

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).