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arXiv · 1301.5672

Class polynomials for nonholomorphic modular functions

Jan Hendrik Bruinier·Ken Ono·Andrew V. Sutherland
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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.

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Jan Hendrik Bruinier, Ken Ono, Andrew V. Sutherland. 2015-01-28. Class polynomials for nonholomorphic modular functions. https://doi.org/10.1016/j.jnt.2015.07.002

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