Class polynomials for nonholomorphic modular functions

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• 作者：Jan Hendrik Bruinier^a ; ^{bruinier@mathematik.tu-darmstadt.de" class="auth_mail" title="E-mail the corresponding author} ; Ken Ono^b ; ^{ono@mathcs.emory.edu" class="auth_mail" title="E-mail the corresponding author} ; Andrew V. Sutherland^c ; ^{drew@math.mit.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Nonholomorphic modular function ; Hilbert class polynomial ; Partition function
• 刊名：Journal of Number Theory
• 出版年：2016
• 出版时间：April 2016
• 年：2016
• 卷：161
• 期：Complete
• 页码：204-229
• 全文大小：524 K

文摘

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 $E_2^{⁎}(z)$, we obtain CRT-based algorithms that compute the class polynomials H_D(F;x)$H_D(F;x)$, whose roots are the discriminant D   singular moduli for F(z)$F(z)$. By applying these results to a specific weak Maass form F_p(z)$F_p(z)$, we obtain a CRT-based algorithm for computing partition class polynomials  , a sequence of polynomials whose traces give the partition numbers p(n)$p(n)$. Under the GRH, the expected running time of this algorithm is O(n^5/2+o(1))$O(n^{5/2+o(1)})$. Key to these results is a fast CRT-based algorithm for computing the classical modular polynomial Φ_m(X,Y)${\mathrm{\Phi }}_m(X,Y)$ that we obtain by extending the isogeny volcano approach previously developed for prime values of m.