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

Keiji Matsumoto, Ryunosuke Nakano

TL;DR

This work connects the iterated mean problem for four terms to an algebro-geometric framework via a period map for a family of fourfold coverings, producing a 3-dimensional ball model $B_3$ embedded into a Siegel space. By constructing four automorphic forms on $B_3$ and establishing their relation to the inverse period map, the authors derive a Jacobi-type theta–period identity and implement a mean-generating transformation that yields recurrence relations among the four means. Central to the approach are Thomae-type formulas linking theta constants to branch points, and a precise description of the inverse period map in terms of Lauricella hypergeometric functions of type $F_D$ in three variables. The main payoff is an explicit expression of the iterated limit in terms of theta constants and Lauricella $F_D$, which unifies Borchardt/Matsumoto–Borwein-type AGM formulas with algebro-geometric data, and recovers Borwein-type results as specializations. The work advances the intersection of iterated means, period maps, and automorphic forms, providing new tools for expressing multidimensional AGM-type limits in terms of classical special functions.

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.

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

TL;DR

This work connects the iterated mean problem for four terms to an algebro-geometric framework via a period map for a family of fourfold coverings, producing a 3-dimensional ball model embedded into a Siegel space. By constructing four automorphic forms on and establishing their relation to the inverse period map, the authors derive a Jacobi-type theta–period identity and implement a mean-generating transformation that yields recurrence relations among the four means. Central to the approach are Thomae-type formulas linking theta constants to branch points, and a precise description of the inverse period map in terms of Lauricella hypergeometric functions of type in three variables. The main payoff is an explicit expression of the iterated limit in terms of theta constants and Lauricella , which unifies Borchardt/Matsumoto–Borwein-type AGM formulas with algebro-geometric data, and recovers Borwein-type results as specializations. The work advances the intersection of iterated means, period maps, and automorphic forms, providing new tools for expressing multidimensional AGM-type limits in terms of classical special functions.

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 embedded into the Siegel upper half-space of degree four. We construct four automorphic forms on 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 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 in three variables.
Paper Structure (20 sections, 208 equations, 3 tables)

This paper contains 20 sections, 208 equations, 3 tables.

Table of Contents

  1. Introduction
  2. A Family of Curves
  3. The configuration space
  4. Fourfold covering $C(x)$ of P1 branching at six points
  5. Holomorphic $1$-Forms and $1$-Cycles
  6. Period matrices
  7. An embedding of $\mathbb{B}_3$ into $\mathfrak{S}_4$
  8. Period maps
  9. Half-turn circuit matrices
  10. Construction of the Inverse of the Period Map
  11. Action of Some Elements of the Symplectic Group on Theta Functions
  12. Relations Between Theta Constants and Branch Points
  13. Actions of Certain Rational Symplectic Elements on Theta Constants
  14. Thomae-Type Formulas
  15. A Mean Generating Transformation
  16. ...and 5 more sections

Theorems & Definitions (34)

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  • ...and 24 more