Class polynomials for nonholomorphic modular functions

TL;DR

This paper develops algorithms to compute class polynomials for nonholomorphic modular functions using isogeny volcanoes and CRT methods, enabling efficient calculation of partition numbers under GRH.

Contribution

It introduces new CRT-based algorithms for class polynomials of nonholomorphic modular functions, extending isogeny volcano techniques and applying them to partition polynomial computations.

Findings

Algorithms for singular moduli of nonholomorphic functions

CRT-based computation of class polynomials under GRH

Fast modular polynomial computation using isogeny volcanoes

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 CRT-based algorithm for computing the classical modular polynomial Phi_m(X,Y) that we obtain by extending the isogeny volcano approach previously developed for prime values of m.