Division polynomials and canonical local heights on hyperelliptic Jacobians

We generalize the division polynomials of elliptic curves to hyperelliptic Jacobians over the complex numbers. We construct them by using the hyperelliptic sigma function. Using the division polynomial, we describe a condition that a point on the Jacobian is a torsion point. We prove several properties of the division polynomials such as determinantal expressions and recurrence formulas. We also study relations among the sigma function, the division polynomials, and the canonical local height functions.