A Maass Lifting of ${\Theta }^3$ and Class Numbers of Real and imaginary Quadratic Fields

Abstract

We give an explicit construct of a harmonic weak Maass form $F_{\Theta }$ that is a "lift" of ${\Theta }^3$, where $\Theta$ is the classical Jacobi theta function. Just as the Fourier coefficients of ${\Theta }^3$ are related to class numbers of imaginary quadratic fields, the Fourier coefficients of the "holomorphic part" of $F_{\Theta }$ are associated to class numbers of real quadratic fields.