Arithmetic relations between modular forms and hypergeometric functions in several variables

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The Schwarz map for a hypergeometric differential equation of 2 variables induces modular forms on the 2-dimensional complex hyperball. By using these modular forms we obtained1) Algebraicity criterion for special values of the Schwarz map for a hypergeometric differential equation,2) Extension in 2 variables of the Gauss arithmetic geometric mean theorem,3) Extension in 2 variables of the Jacobi theta recursive theorem,4) Explicit expression of the Shimura curve with discriminant 6.