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Division polynomials in Mumford coordinates

Julia Bernatska

TL;DR

This work develops an algebraic framework to compute division polynomials for genus-two hyperelliptic curves using Mumford coordinates, enabling direct construction of $n$-torsion divisors from these polynomials. By formulating addition and duplication laws within the polynomial-function ring $oldsymbol{ ext P}( ext{C})$ and solving the Jacobi inversion problem via $oldsymbol{ ext{R}}_4$ and $oldsymbol{ ext{R}}_5$, it provides explicit $3$- and $4$-torsion polynomials and the corresponding $x$- and $y$-coordinates of torsion divisor supports. The approach avoids heavy reliance on $ ext{wp}$-function identities, instead leveraging Mumford coordinates and determinant constructions to yield compact, verifiable expressions, with Mathematica implementations confirming the results. The method generalizes to higher genus and non-hyperelliptic curves, offering a practical tool for isogeny-based cryptography where torsion structures on Jacobians are essential. Overall, the paper delivers a concrete algebraic pathway from division polynomials to torsion divisors in genus-two settings, with clear recipes for both computation and verification via Jacobi inversion.

Abstract

An effective method of computing division polynomials in terms of Mumford coordinates is presented. As an example, division polynomials for $3$- and $4$-torsion divisors on a genus two curve are obtained explicitly in terms of Mumford coordinates, and $x$-, $y$-coordinates of the support of torsion divisors. As a result, $n$-torsion divisors on a given curve can be computed directly from the division polynomials. Alternatively, these divisors are obtained by solving the Jacobi inversion problem at points of the Jacobian variety of order $n$.

Division polynomials in Mumford coordinates

TL;DR

This work develops an algebraic framework to compute division polynomials for genus-two hyperelliptic curves using Mumford coordinates, enabling direct construction of -torsion divisors from these polynomials. By formulating addition and duplication laws within the polynomial-function ring and solving the Jacobi inversion problem via and , it provides explicit - and -torsion polynomials and the corresponding - and -coordinates of torsion divisor supports. The approach avoids heavy reliance on -function identities, instead leveraging Mumford coordinates and determinant constructions to yield compact, verifiable expressions, with Mathematica implementations confirming the results. The method generalizes to higher genus and non-hyperelliptic curves, offering a practical tool for isogeny-based cryptography where torsion structures on Jacobians are essential. Overall, the paper delivers a concrete algebraic pathway from division polynomials to torsion divisors in genus-two settings, with clear recipes for both computation and verification via Jacobi inversion.

Abstract

An effective method of computing division polynomials in terms of Mumford coordinates is presented. As an example, division polynomials for - and -torsion divisors on a genus two curve are obtained explicitly in terms of Mumford coordinates, and -, -coordinates of the support of torsion divisors. As a result, -torsion divisors on a given curve can be computed directly from the division polynomials. Alternatively, these divisors are obtained by solving the Jacobi inversion problem at points of the Jacobian variety of order .

Paper Structure

This paper contains 17 sections, 14 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hyperelliptic curve of genus two
  4. Entire functions
  5. Uniformization of the curve
  6. Polynomial functions on a curve
  7. Addition and duplication
  8. Addition law
  9. Addition law on special divisors
  10. Duplication law
  11. The case of $D_4 \sim D_1$
  12. Torsion divisors
  13. $2$-Torsion divisors
  14. $3$-Torsion divisors
  15. $4$-Torsion divisors
  16. ...and 2 more sections

Key Result

Proposition 1

bakerAF Given $u\in \mathrm{Jac}(\mathcal{C})\backslash (\sigma)_0$, the Abel pre-image of $u$ is a non-special divisor $D_2 = (x_1,y_1)+(x_2,y_2)$ with coordinates uniquely defined by the system where $u = \mathcal{A}(D_2)$, In other words, $D_2$ is a common divisor of zeros of the two polynomial functions $\mathcal{R}_{4}$, $\mathcal{R}_{5}$ on the curve $\mathcal{C}$.

Theorems & Definitions (25)

  • Proposition 1: The Jacobi inversion problem
  • Proposition 2
  • proof
  • Remark 1
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • proof
  • Corollary 1
  • proof
  • ...and 15 more