Maass lifts of half-integral weight Eisenstein series and theta powers

TL;DR

This paper constructs explicit mock modular forms with Eisenstein series shadows of various weights and levels, and provides harmonic weak Maass forms related to theta powers, advancing understanding of modular objects and their interrelations.

Contribution

It introduces explicit constructions of mock modular forms with specified Eisenstein series shadows and harmonic Maass forms linked to theta powers, expanding the toolkit for studying modular forms.

Findings

Explicit mock modular forms with Eisenstein series shadows constructed.

Harmonic weak Maass forms as preimages of theta powers obtained.

New connections between Eisenstein series, Maass forms, and theta functions established.

Abstract

In this paper, we explicitly construct mock modular forms whose shadows are Eisenstein series of arbitrary integral and half-integral weight, level and character at the cusps $\infty$ and $0$. As an application, we give explicit construction of harmonic weak Maass forms which are Hecke eigenforms and are the preimages of $\Theta^k, k \in \{ 3, 5, 7\}$ under the shadow operator, where $\Theta$ is the classical Jacobi theta function.