Modular polynomials via isogeny volcanoes

TL;DR

This paper introduces a new algorithm leveraging isogeny volcanoes to efficiently compute classical modular polynomials for large n and m, significantly advancing computational capabilities in this area.

Contribution

The paper presents a novel algorithm that uses the graph of n-isogenies and the CRT to compute modular polynomials more efficiently than previous methods.

Findings

Successfully computed Phi_n for n over 5000

Computed Phi_n mod m for n over 20000

Handled larger n values using modular functions g

Abstract

We present a new algorithm to compute the classical modular polynomial Phi_n in the rings Z[X,Y] and (Z/mZ)[X,Y], for a prime n and any positive integer m. Our approach uses the graph of n-isogenies to efficiently compute Phi_n mod p for many primes p of a suitable form, and then applies the Chinese Remainder Theorem (CRT). Under the Generalized Riemann Hypothesis (GRH), we achieve an expected running time of O(n^3 (log n)^3 log log n), and compute Phi_n mod m using O(n^2 (log n)^2 + n^2 log m) space. We have used the new algorithm to compute Phi_n with n over 5000, and Phi_n mod m with n over 20000. We also consider several modular functions g for which Phi_n^g is smaller than Phi_n, allowing us to handle n over 60000.