Class polynomials for nonholomorphic modular functions

Abstract

Abstract. We give algorithms for computing the singular moduli of suitable nonholomorphic modular functions F (z). By combining the theory of isogeny volcanoes with a beautiful observation of Masser concerning the nonholomorphic Eisenstein series E * 2 (z), we obtain CRT-based algorithms that compute the class polynomials H D (F ; x), whose roots are the discriminant D singular moduli for F (z). By applying these results to a specific weak Maass form F p (z), we obtain a CRT-based algorithm for computing partition class polynomials, a sequence of polynomials whose traces give the partition numbers p(n). Under the GRH, the expected running time of this algorithm is O(n 5/2+o(1) ). Key to these results is a fast CRTbased algorithm for computing the classical modular polynomial Φ m (X, Y ) that we obtain by extending the isogeny volcano approach previously developed for prime values of m.

On Witten’s Extremal Partition Functions

“…Remark. The results in [6] indicate how to efficiently compute p(n) as traces of singular moduli numerically, and may be useful to those wishing to implement the identities presented here.…”

Modular invariants and isogenies

“…In Corollary 4.3, we prove that for all m ≥ 1 (6) h ∞ (Φ m ) ≤ ψ(m) 6 log m + log ψ(m) + 6 log(12 log m + 2 log ψ(m) + 25.2) + 15.7 , so the main term in the upper bound is slightly worse 1 than the known asymptotic, recalled in (55), when m grows to infinity. Estimates on modular polynomials are used for instance when computing explicitly Hilbert class polynomials, see for instance [9,5].…”

Towards Practical Key Exchange from Ordinary Isogeny Graphs

“…This subgroup is the kernel of a separable isogeny φ a . 9 The codomain E/E[a] of φ a is well-defined up to isomorphism and will be denoted a · E. The isogeny φ a is always horizontal-that is, End(a · E) = End(E)-and its degree is the norm of a as an ideal of End(E).…”

“…An order is a subring which is a Z-module of rank 2. 8 ∆K is a fundamental discriminant: ∆K ≡ 0, 1 (mod 4), and ∆K or ∆ K 4 is squarefree 9. In fact, one can define φa for any invertible ideal a, but it is not always separable.…”