Computing isogenies between Jacobian of curves of genus 2 and 3

TL;DR

This paper introduces a quasi-linear algorithm for computing isogenies between Jacobians of genus 2 and 3 curves, extending previous methods and including non-hyperelliptic cases using algebraic theta functions.

Contribution

It generalizes existing algorithms for genus 2 to genus 3 curves and handles non-hyperelliptic cases with new algebraic theta function techniques.

Findings

Algorithm operates in quasi-linear time

Successfully extends to genus 3 hyperelliptic and non-hyperelliptic curves

Improves upon previous genus 2 isogeny computation methods

Abstract

We present a quasi-linear algorithm to compute isogenies between Jacobians of curves of genus 2 and 3 starting from the equation of the curve and a maximal isotropic subgroup of the l-torsion, for l an odd prime number, generalizing the V\'elu's formula of genus 1. This work is based from the paper "Computing functions on Jacobians and their quotients" of Jean-Marc Couveignes and Tony Ezome. We improve their genus 2 case algorithm, generalize it for genus 3 hyperelliptic curves and introduce a way to deal with the genus 3 non-hyperelliptic case, using algebraic theta functions.