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

TL;DR

This paper surveys an algorithm for efficiently computing explicit isogenies between Jacobians of hyperelliptic curves of any genus over p-adic fields, with quasi-linear complexity in the degree and genus.

Contribution

It introduces a quasi-linear complexity algorithm for computing rational representations of isogenies between hyperelliptic Jacobians of arbitrary genus.

Findings

Algorithm has quasi-linear complexity in degree and genus

Applicable over extensions of p-adic fields

Enables explicit isogeny computations for hyperelliptic Jacobians

Abstract

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