The algebro-geometric aspect of the iterated limit of a quaternary of means of four terms

Abstract

We study the iterated limit of a quaternary of means of four terms through the period map from the family of cyclic fourfold coverings of the complex projective line branching at six points to the three-dimensional complex ball $\mathbb{B}_3$ embedded into the Siegel upper half-space of degree four. We construct four automorphic forms on $\mathbb{B}_3$ expressing the inverse of the period map, and give an equality between one of them and a period integral, which is an analogy of Jacobi's formula between a theta constant and an elliptic integral. We find a transformation of $\mathbb{B}_3$ such that the quaternary of means appears by its actions on the four automorphic forms. These results enable us to express the iterated limit by the Lauricella hypergeometric series of type $D$ in three variables.