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## Section: New Results

### Explicit Isogenies in Genus 2 and 3

Participant : Enea Milio.

In [22], 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 $\ell$-torsion, for $\ell$ an odd prime number, generalizing Vélu’s formula of genus 1. This work is based on 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.