  Let $p$ be an odd prime number. We propose an algorithm for computing
rational representations of isogenies between Jacobians of hyperelliptic curves
via-adic differential equations with a sharp analysis of the loss of precision.
Consequently, after having possibly lifted the problem in the $p$-adics, we
derive fast algorithms for computing explicitly Cantor's division polynomials
of hyperelliptic curves defined over finite fields.
