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

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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.