[Paper Review] About the group law for the Jacobi variety of a hyperelliptic curve

Frank Leitenberger|ArXiv.org|Sep 27, 2005

Algebraic Geometry and Number Theory10 references4 citations

TL;DR

This paper presents explicit rational formulas for the group law on the Jacobian of a genus 2 hyperelliptic curve using cubic interpolation polynomials, generalizing the chord-and-tangent method for elliptic curves. The key contribution is a one-step reduction via a cubic parabola that yields closed-form rational expressions for addition in the Jacobian, valid on a dense open subset, avoiding iterative algorithms and root operations.

ABSTRACT

We generalize the group law of curves of degree three by chords and tangents to the Jacobi variety of a hyperelliptic curve. In the case of genus 2 we accomplish the construction by a cubic parabola. We derive explicit rational formulas for the addition on a dense set in the Jacobian.

Motivation & Objective

• To provide an explicit, closed-form description of the group law on the Jacobian of a hyperelliptic curve of genus 2.
• To generalize the classical chord-and-tangent construction for elliptic curves to higher genus via interpolation.
• To derive rational formulas for addition in the Jacobian that avoid iterative reduction steps and root operations.
• To offer a geometric and algebraic framework for the group law using interpolation functions inspired by Jacobi and Abel.
• To resolve the lack of explicit formulas in the literature, as noted by Cassels and Mazur, for genus 2 Jacobians.

Proposed method

• Construct a cubic interpolation polynomial through the x-coordinates and y-values of two reduced divisors in the Jacobian.
• Use the intersection of this cubic with the hyperelliptic curve to find additional points, whose conjugates yield the sum in the Jacobian.
• Apply Vieta's formulas to compute the sum of x-coordinates of the new intersection points directly from coefficients of the cubic and curve.
• Express the result in terms of Mumford coordinates (A(x), B(x)) to achieve rational group law formulas.
• Use Groebner basis computations to eliminate the original points and express the group law coefficients solely in terms of Mumford coordinates.
• Derive explicit rational expressions for α₃, β₃, γ₃, δ₃ in terms of α₁, β₁, γ₁, δ₁, α₂, β₂, γ₂, δ₂, and curve coefficients, avoiding radicals.

Experimental results

Research questions

• RQ1 Can the group law on the Jacobian of a genus 2 hyperelliptic curve be expressed by explicit rational formulas without iterative reduction steps?
• RQ2 How can the classical chord-and-tangent method for elliptic curves be generalized to higher genus using interpolation?
• RQ3 What is the role of cubic interpolation polynomials in achieving a one-step reduction of divisor sums in the Jacobian?
• RQ4 Can the group law be expressed rationally in Mumford coordinates without involving square roots or resultants?
• RQ5 How do the geometric properties of the curve and its Jacobian relate to the algebraic structure of the group law?

Key findings

• The paper derives explicit rational formulas for the group law on the Jacobian of a genus 2 hyperelliptic curve, valid on a dense open subset of the Jacobian minus the theta divisor.
• The sum of two divisors is computed in a single step using a cubic interpolation polynomial through the points, with the resulting intersections determining the sum via their conjugates.
• The formulas for α₃, β₃, γ₃, δ₃ are rational functions of the Mumford coordinates of the inputs and the curve coefficients, avoiding radicals.
• The derivation uses Groebner basis techniques to eliminate original points and express the group law coefficients solely in terms of Mumford data.
• The method generalizes to higher genus using rational interpolation functions of appropriate degree, with the genus 2 case being the most explicit due to the cubic nature of the interpolation.
• The formulas remain valid in the limit of repeated points (e.g., doubling), extending to singular cases via continuity.