Class polynomials for nonholomorphic modular functions

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 E2⁎(z), we obtain CRT-based algorithms that compute the class polynomials HD(F;x), whose roots are the discriminant D singular moduli for F(z). By applying these results to a specific weak Maass form Fp(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(n5/2+o(1)). Key to these results is a fast CRT-based algorithm for computing the classical modular polynomial Φm(X,Y) that we obtain by extending the isogeny volcano approach previously developed for prime values of m.