Modular Polynomials Via Isogeny Volcanoes

We present a new algorithm to compute the classical modular polynomial Phi(l) in the rings Z[X, Y] and (Z/mZ)[X, Y], for a prime l and any positive integer m. Our approach uses the graph of l-isogenies to efficiently compute Phi(l) 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(l(3)(log l)(3) log log l), and compute Phi(l) mod m using O(l(2)(log l)(2) + l(2) log m) space. We have used the new algorithm to compute Phi(l) with l over 5000, and Phi(l) mod m with l over 20000. We also consider several modular functions g for which Phi(g)(l) is smaller than Phi(l), allowing us to handle l over 60000.