[Paper Review] Determinant Expressions for Hyperelliptic Functions (with an Appendix by Shigeki Matsutani)

Yoshihiro Ônishi|ArXiv.org|May 23, 2001

Algebraic Geometry and Number Theory11 references19 citations

TL;DR

This paper generalizes classical determinant identities for elliptic functions to hyperelliptic curves of arbitrary genus, expressing special combinations of $σ$-functions and their derivatives as determinants of matrices built from derivatives of $x(u)$ and $y(u)$, the coordinates on the curve. The key contribution is a unified determinant formula that recovers and extends Fay’s formula and classical identities for elliptic functions, with a novel approach focusing on the curve rather than its Jacobian.

ABSTRACT

In this paper we give quite pretty generalization of the formula of Frobenius-Stickelberger to all hyperelliptic curves. The formula of Kiepert type is also obtained by limiting process from this generalization. In Appendix a determinant expression of D.G. Cantor is also given.

Motivation & Objective

• To generalize classical determinant formulas for elliptic functions to hyperelliptic curves of arbitrary genus.
• To provide a unified determinant expression for combinations of $σ$-functions and their derivatives that extends both Fay’s formula and classical identities.
• To shift focus from the Jacobian variety to the curve itself, as inspired by Grant’s philosophy, for a more natural generalization.
• To establish a determinant formula that naturally generalizes both (0.3) and (0.4) from the introduction, valid for all hyperelliptic curves.

Proposed method

• Define hyperelliptic functions as lifts to the universal abelian covering of the curve, using the Abel map to identify points on the curve with vectors in $\mathbb{C}^g$.
• Construct a determinant matrix whose entries are derivatives of monomials $x^k$ and $yx^k$ in the coordinate functions $x(u)$ and $y(u)$ on the curve.
• Use differential operators $\left(\frac{d}{dx}\right)^k$ with binomial coefficients to build the matrix entries, reflecting the structure of Taylor expansions.
• Apply a column reordering and scaling to transform the determinant into a form involving factorials and derivatives of $y(u)$, leading to a clean determinant expression.
• Leverage the structure of the matrix to factor out terms like $1!2!\cdots(r!)$ and relate the determinant to a product of $\sigma$-functions via a change of basis.
• Verify the formula by comparing with known identities in genus 1 and 2, and prove the general case using induction and combinatorial identities in the appendix.

Experimental results

Research questions

• RQ1 Can classical determinant identities for elliptic functions be generalized to hyperelliptic curves of arbitrary genus?
• RQ2 Is there a determinant formula that unifies Fay’s formula and the classical identities (0.3) and (0.4) for higher-genus curves?
• RQ3 Can such a generalization be achieved by focusing on the curve itself rather than its Jacobian variety?
• RQ4 What is the precise structure of the determinant matrix that captures the behavior of $σ$-functions on hyperelliptic curves?
• RQ5 How do the combinatorial coefficients and derivatives of $x(u)$ and $y(u)$ combine to yield a clean expression for $\sigma(nu)/\sigma(u)^{n^2}$?

Key findings

• The paper establishes a general determinant formula for $\sigma(nu)/\sigma(u)^{n^2}$ in terms of derivatives of $x(u)$ and $y(u)$, valid for any hyperelliptic curve of genus $g \geq 1$.
• The determinant expression matches the classical identities (0.3) and (0.4) in genus 1, and generalizes them to all genera.
• The formula is derived by reordering and scaling columns of a matrix built from $x^k$ and $yx^k$ derivatives, with coefficients involving binomial and factorial terms.
• The final determinant form is equivalent to a matrix of differential operators acting on $y(u)$, scaled by factorials, yielding a clean expression involving $y^{(k)}(u)/k!$.
• The derivation confirms that the sign factor $(-1)^{n-1}$ and the factorial product $1!2!\cdots(n-1)!$ in the classical case are naturally reproduced in the general setting.
• The appendix provides a complete proof of the determinant identity (Theorem A.1) through column manipulation and combinatorial identities, validating the main result.