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

TL;DR

This paper explores the algebraic geometry of iterated means of four terms using period maps, automorphic forms, and hypergeometric series, revealing deep connections between these mathematical structures.

Contribution

It constructs automorphic forms related to the period map of cyclic fourfold coverings and links the iterated limit of means to Lauricella hypergeometric series.

Findings

Constructed four automorphic forms expressing the inverse of the period map.

Established an equality between an automorphic form and a period integral.

Expressed the iterated limit using Lauricella hypergeometric series of type D.

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.