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

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 [Formula: see text] embedded into the Siegel upper half-space of degree four. We construct four automorphic forms on [Formula: see text] expressing the inverse of the period map, and give an equality between one of them and a period integral, which is an analogue of Jacobi’s formula between a theta constant and an elliptic integral. We find a transformation of [Formula: see text] such that the quaternary of means appears through its action 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.