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

TL;DR

This paper constructs a harmonic weak Maass form lifting the cube of the Jacobi theta function, linking its Fourier coefficients to class numbers of both real and imaginary quadratic fields, thus bridging different areas of number theory.

Contribution

It provides an explicit construction of a harmonic weak Maass form that lifts  and connects class numbers of real and imaginary quadratic fields.

Findings

Fourier coefficients of the holomorphic part relate to real quadratic class numbers.

Explicit construction of a Maass form lifting .

Establishes a new link between Maass forms and quadratic field class numbers.

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.