Computing isogenies between jacobians of hyperelliptic curves of arbitrary genus via differential equations

Abstract

Let p be an odd prime number and ℓ be an integer coprime to p. We survey an algorithm for computing explicit rational representations of (ℓ, . . . , ℓ)-isogenies between Jacobians of hyperelliptic curves of arbitrary genus over an extension K of the field of p-adic numbers Qp. The algorithm has a quasi-linear complexity in ℓ as well as in the genus of the curves.