Modular polynomials via isogeny volcanoes

We present a new algorithm to compute the classical modular polynomial Φl 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 Φl 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(l3(log l)3 log log l), and compute Φl mod m using O(l 2(log l)2+ l2 logm) space. We have used the new algorithm to compute Φl with l over 5000, and Φl mod m with l over 20000. We also consider several modular functions g for which Φ l is smaller than Φl, allowing us to handle l over 60000.