Division polynomials in Mumford coordinates

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 $n$.