Division polynomials in Mumford coordinates

TL;DR

This paper introduces an effective method to compute division polynomials using Mumford coordinates, enabling explicit calculations of torsion divisors on genus two curves.

Contribution

It provides a novel approach to compute division polynomials in Mumford coordinates and explicitly derives torsion divisors for genus two curves.

Findings

Division polynomials for 3- and 4-torsion divisors are explicitly obtained.

n-torsion divisors can be computed directly from division polynomials.

Alternative method involves solving the Jacobi inversion problem.

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 $n$.