Canonical heights and division polynomials (Q2927890)

An efficient algorithm for computing the canonical height of an algebraic point on an elliptic curve \(E\) over a number field has been devised by \textit{G. Everest} and \textit{T. B. Ward} [New York J. Math. 6, 331--342 (2000; Zbl 0973.11062)]. It relies on the division polynomial \(\phi_n\) associated to \(E\) and on a Diophantine approximation result. The authors extend this approach to hyperelliptic Jacobians over a number field. The method involves recurrence relations obtained by \textit{Y. Uchida} [Manuscr. Math. 134, No. 3--4, 273--308 (2011; Zbl 1226.14039)] for the analog of \(\phi_n\) for Jacobians of hyperelliptic curves. It also involves a Diophantine approximation result due to \textit{G. Faltings} [Ann. Math. (2) 133, No. 3, 549--576 (1991; Zbl 0734.14007)]. The authors have implemented the computation of the values of \(\phi_n\) in the case of genus \(2\). They give explicit examples of height computation.