Modular polynomials via isogeny volcanoes

Abstract

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

Isogeny graphs of ordinary abelian varieties

“…1. This description has seen numerous applications, including point-counting on elliptic curves [2], random self-reducibility of the elliptic curve discrete logarithm problem in isogeny classes [3,4], generating elliptic curves with a prescribed number of points via the CM method [5], and computing modular polynomials [6].…”

An explicit height bound for the classical modular polynomial

“…A direct computation, using the bound B 1 (l) and the algorithm described in [6], finds that Theorem 1 holds for l < 3600 (see Table 1). For all l ≥ 3600, the bound of the theorem is greater than B 2 (l), hence the Theorem 1 holds for l ≥ 3600.…”

“…Such an explicit bound is needed to obtain rigorous results from algorithms that compute Φ l , including [6,7,10,11]. To prove Theorem 1, we retrace the proof of Cohen, specializing to the case that m = l is prime and seeking only an upper bound.…”

Computations with Witt vectors of length 3

“…Note that the proof of item 2 allows us to reformulate the second item of Conjecture 9.3 in the following way: The pattern from this conjecture was observed by the author using the formulas for the modular polynomials Φ p (X, Y ) for p ≤ 353 computed by M. Rubinstein, which are available at http://www.math.uwaterloo.ca/~mrubinst/modularpolynomials/ phi_l.html Later, A. V. Sutherland, using methods from [1], was able to verify it for p < 2500. (He also observed that his methods would actually allow him go much further.…”