Theta lifts and local Maass forms

TL;DR

This paper introduces a new class of modular objects called locally harmonic Maass forms, realized as theta lifts of harmonic weak Maass forms, and constructs examples of local Maass forms as eigenfunctions of the hyperbolic Laplacian.

Contribution

It establishes a connection between locally harmonic Maass forms and theta lifts, providing explicit constructions of local Maass forms as Laplacian eigenfunctions.

Findings

Locally harmonic Maass forms can be realized as theta lifts.

Constructed examples of non-harmonic local Maass forms.

Demonstrated that these forms are eigenfunctions of the hyperbolic Laplacian.

Abstract

The first two authors and Kohnen have recently introduced a new class of modular objects called locally harmonic Maass forms, which are annihilated almost everywhere by the hyperbolic Laplacian operator. In this paper, we realize these locally harmonic Maass forms as theta lifts of harmonic weak Maass forms. Using the theory of theta lifts, we then construct examples of (non-harmonic) local Maass forms, which are instead eigenfunctions of the hyperbolic Laplacian almost everywhere.