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Mathematics of Computation

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Computing isogenies between Jacobians of curves of genus 2 and 3

Author: Enea Milio
Journal: Math. Comp. 89 (2020), 1331-1364
MSC (2010): Primary 14K02, 14K25, 14Q05, 14Q20
Published electronically: October 28, 2019
MathSciNet review: 4063320
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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.

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Additional Information

Enea Milio
Affiliation: chemin des Peupliers 4, 1028 Préverenges, Switzerland

Received by editor(s): October 9, 2018
Received by editor(s) in revised form: July 20, 2019, and August 7, 2019
Published electronically: October 28, 2019
Article copyright: © Copyright 2019 American Mathematical Society