Systems of uniformization equations and hyperelliptic integrals

Abstract

The purpose of this paper is to study systems of uniformization equations with respect to Saito free divisors which have solutions expressed in terms of hyperelliptic integrals. There are two such divisors. Both are hypersurfaces in a three-dimensional affine space defined by weighted homogeneous polynomials. One is constructed by the discriminant of a dihedral group of order 2(2n+1). The other is the discriminant of the reflection group of type H3. In the former case, we construct fundamental solutions by Gaussian hypergeometric functions in addition to a solution expressed by a hyperelliptic integral.