Division polynomials and canonical local heights on hyperelliptic Jacobians (Q625494)

The notion of a division polynomial is well-known for elliptic curves. Recently, several authors have given generalizations to the case of hyperelliptic curves and hyperelliptic Jacobians. The former case has been studied algebraically in [\textit{D. G. Cantor}, J. Reine Angew. Math. 447, 91--145 (1994; Zbl 0788.14026)] and through the hyperelliptic sigma function in a series of paper by Ônishi. The latter case is worked out in [\textit{N. Kanayama}, Math. Proc. Camb. Philos. Soc. 139, No. 3, 399-409 (2005); corrections ibid. 149, No. 1, 189--192 (2010; Zbl 1093.14042)] for genus \(2\). In this article, the author generalizes this last approach to any genus. He gives a determinantal expression of the division polynomial in terms of the hyperelliptic \(\wp\)-functions and also recurrence formulae. He then describes a condition that a point on the Jacobian is a torsion point and study relations with local height functions.