[Paper Review] Degenerations of triple coverings and Thomae's formula

Keiji Matsumoto, Tomohide Terasoma|arXiv (Cornell University)|Jan 27, 2010

Advanced Algebra and Geometry8 references3 citations

TL;DR

This paper proves Thomae’s formula for triple coverings of the complex projective line by constructing symplectic bases using binary tree combinatorics, and computes the absolute constant in the formula for arbitrary branching indices. The key contribution is a closed-form expression for the constant via degeneration techniques and a variant of the Chowla-Selberg formula, validated through stable degenerations into unions of two triple coverings.

ABSTRACT

In this paper, we prove Thomae's formula for a triple covering of $\bold P^1$ with arbitrary index. This formula gives a relation between theta constants, determinants of period integrals and the difference products of branch points. To specify a symplectic basis of the curve, we use the combinatorics of binary trees on $\bold P^1$. This symplectic basis behaves so well for degenerations that we obtain the absolute constant in this formula and reduce it to a special case treated in [Bershadsky-Radul], [Nakayashiki].

Motivation & Objective

• To establish Thomae’s formula for triple coverings of P¹ with explicit absolute constants for arbitrary branching indices.
• To define a canonical symplectic basis for the homology of triple coverings using planar binary trees.
• To analyze degenerations of triple coverings into unions of two irreducible components via tree decomposition.
• To compute the absolute constant in Thomae’s formula using stable degenerations and the Chowla-Selberg-type formula.

Proposed method

• Construct symplectic bases of the homology group H₁(C, ℤ) from planar binary trees on P¹, ensuring compatibility under degeneration.
• Use degeneration of the curve to a union of two triple coverings, corresponding to a decomposition of the binary tree into two subtrees.
• Apply the Chowla-Selberg-type formula (equation 5.2) to relate the limit of period matrices and difference products to the theta constants.
• Leverage the monodromy-invariant degeneration to reduce the general case to a known special case via recursive degeneration steps.
• Use the automorphism ρ: (x,y) ↦ (x,ωy) to analyze the (1−ρ)-torsion part of H₁(C, ℚ/ℤ) and relate it to theta characteristics.
• Verify the formula by induction on the tree decomposition, using the limit of theta constants at degenerate fibers to derive the absolute constant.

Experimental results

Research questions

• RQ1 How can Thomae’s formula for triple coverings be extended to arbitrary branching indices with explicit absolute constants?
• RQ2 What is the role of binary tree combinatorics in defining symplectic bases that behave well under degeneration?
• RQ3 How do degenerations of triple coverings into unions of two components affect the symplectic basis and theta constants?
• RQ4 Can the absolute constant in Thomae’s formula be computed explicitly using degeneration and special function identities?
• RQ5 What is the precise dependence of the constant on the choice of symplectic basis and theta characteristic?

Key findings

• The absolute constant in Thomae’s formula for triple coverings is given by κ_Λ = ±(2π)^3 3^{3/4} exp(11πi/12) κ_Λ′, derived via degeneration and special function identities.
• The symplectic basis constructed from a binary tree extends continuously to degenerate fibers, decomposing into bases for two irreducible components upon tree splitting.
• The formula reduces to a known case via recursive degeneration, enabling explicit computation of the constant through limits of period matrices and difference products.
• The sixth power of the theta constant is proportional to the product of the determinant of the period matrix cubed and the difference product of branch points, with an explicit constant factor.
• For the example with four real branch points, the constant is explicitly computed as 1/(3√3 (2π)^6) exp(πi/6) times the product of differences and det(B)^3.
• The method avoids reliance on prior generalizations and instead uses degeneration and tree-based symplectic bases to derive the constant directly.