Class polynomials for nonholomorphic modular functions

We give algorithms for computing the singular moduli   of suitable nonholomorphic modular functions F(z)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)HD(F;x), whose roots are the discriminant D   singular moduli for F(z)F(z). By applying these results to a specific weak Maass form Fp(z)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)p(n). Under the GRH, the expected running time of this algorithm is O(n5/2+o(1))O(n5/2+o(1)). Key to these results is a fast CRT-based algorithm for computing the classical modular polynomial Φm(X,Y)Φm(X,Y) that we obtain by extending the isogeny volcano approach previously developed for prime values of m.