Computing isogenies between Jacobians of curves of genus 2 and 3

摘要

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}}o 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.