Computing isogenies between Jacobians of curves of genus 2 and 3

Abstract:

We present a quasi-linear algorithm to compute (separable) isogenies of degree $\ell ^g$, for $\ell$ an odd prime number, between Jacobians of curves of genus $g=2$ and $3$ starting from the equation of the curve $\mathcal {C}$ and a maximal isotropic subgroup $\mathcal {V}$ of the $\ell$-torsion, generalizing Vélu’s formula from genus $1$. Denoting by $J_{\mathcal {C}}$ the Jacobian of $\mathcal {C}$, the isogeny is $J_{\mathcal {C}}\to J_{\mathcal {C}}/\mathcal {V}$. Thus $\mathcal {V}$ is the kernel of the isogeny and we compute only isogenies with such kernels. 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 $2$ algorithm, generalize it to genus $3$ hyperelliptic curves, and introduce a way to deal with the genus $3$ nonhyperelliptic case, using algebraic theta functions.