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

TL;DR

This paper proves a general theorem confirming that certain CM-values of weak Maass forms are algebraic integers, extending previous results on harmonic Maass forms and their coefficients related to partition functions.

Contribution

The paper establishes a broad theorem demonstrating the algebraic integrality of CM-values of weak Maass forms, confirming a conjecture related to harmonic Maass form coefficients.

Findings

Proves that (24n-1) P(α_Q) is an algebraic integer.

Extends Bruinier and Ono's results to a general setting.

Confirms a conjecture about algebraic integrality of CM-values.

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 \[p(n)=\frac{1}{24n-1} \sum P(\alpha_Q),\] 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.