Integrality Properties of the CM-values of Certain Weak Maass Forms

Abstract

In a recent paper, Bruinier and Ono prove that the coefficients of certain weight -1/2 harmonic Maass forms are traces of singular moduli for weak Maass forms. In particular, for the partition function $p(n)$, they prove that \[p(n)=\frac{1}{24n-1} \sum P(α_Q),\] where $P$ is a weak Maass form and ${\alpha }_Q$ ranges over a finite set of discriminant $-24n+1$ CM points. Moreover, they show that $6(24n-1)P({\alpha }_Q)$ is always an algebraic integer, and they conjecture that $(24n-1)P({\alpha }_Q)$ is always an algebraic integer. Here we prove a general theorem which implies this conjecture as a corollary.