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arXiv · 2408.06990

Computing modular polynomials by deformation

Sabrina Kunzweiler·Damien Robert
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Abstract

We present an unconditional CRT algorithm to compute the modular polynomial $\Phi_\ell(X,Y)$ in quasi-linear time. The main ingredients of our algorithm are: the embedding of $\ell$-isogenies in smooth-degree isogenies in higher dimension, and the computation of $m$-th order deformations of isogenies. We provide a proof-of-concept implementation of a heuristic version of the algorithm demonstrating the practicality of our approach. Our algorithm can also be used to compute the reduction of $\Phi_{\ell}$ modulo $p$ in quasi-linear time (with respect to $\ell$) $\tilde{O}(\ell^2(\log p + \log \ell)^{\mathfrak{O}})$.

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Sabrina Kunzweiler, Damien Robert. 2024-08-13. Computing modular polynomials by deformation. https://arxiv.org/abs/2408.06990

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