Studies on the hypergeometric diffeential equations considering the application for the coding theory

在本课题中，我们得到了如下结果：一般情况下，我们没有关于高斯超几何微分方程解的Schwarz映射的代数值。所以我们被要求分析这个问题，我们什么时候有它的代数值？这是一个难题，许多数学家多年来一直在研究它。通过与法兰克福的J.Wolfart教授的研究，我们发现这个问题与CM场在微分空间上的作用密切相关。结果发表在(1)和(2)中。2001年，我们研究了具有6个参数的K3曲面族，其特征在于Picard格子的结构。在前人研究成果的基础上，我们进行了一些算术应用的研究，并对K3曲面族中与自反多面体相关的模函数的进一步研究进行了规划。

We obtained the following results in our project.In general we don't have algebraic values for the Schwarz map of Gauss hypergeometric differential equations foir tan algebraic argument. So we are asked to ansew the question, when we have its algebraic value? It is a difficult problem, and has been studied for many years by many mathematicians. By the study together with professor J.Wolfart in Frankfurt we discovered the problem is closely connected with the action of the CM field on the space of differentials. The results are pubished in (1)and (2).In 2001 we made a study about the family with 6 parameters of K3 surfaces characterized by the structure of the Picard lattice. We make a research program to obtain some arithmetic application based on the these previous results.We made a plan of the future research of modular functions induced from the family of K3 surfaces related to reflexive polytopes.