Integrality properties of the CM-values of certain weak Maass forms

Abstract

Abstract In a recent paper, Bruinier and Ono prove that the coefficients of certain weight - 1 / 2 $-1/2$ 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 ) = 1 24 n - 1 · ∑ P p ( α Q ) , $ p(n)=\frac{1}{24n-1}\cdot \sum P_p(\alpha _Q), $ where Pp is a weak Maass form and α Q $\alpha _Q$ ranges over a finite set of discriminant - 24 n + 1 $-24n+1$ CM points. Moreover, they show that 6 · ( 24 n - 1 ) · P p ( α Q ) $6\cdot (24n-1)\cdot P_p(\alpha _Q)$ is always an algebraic integer, and they conjecture that ( 24 n - 1 ) · P p ( α Q ) $(24n-1)\cdot P_p(\alpha _Q)$ is always an algebraic integer. Here we prove a general theorem which implies this conjecture as a corollary.