Borcherds lifts of harmonic Maass forms and modular integrals

TL;DR

This paper extends Borcherds' singular theta lift to harmonic Maass forms with exponential growth, analyzes its singularities, and constructs automorphic products, broadening the understanding of modular integrals and automorphic forms.

Contribution

It generalizes Borcherds' lift to a wider class of harmonic Maass forms and constructs new automorphic products, advancing the theory of modular forms and their lifts.

Findings

The lift is continuous but not differentiable along certain geodesics.

Explicit Fourier expansions of the lift are obtained.

A higher-level generalization of a known modular integral is achieved.

Abstract

We extend Borcherds' singular theta lift in signature $(1,2)$ to harmonic Maass forms of weight $1/2$ whose non-holomorphic part is allowed to be of exponential growth at $i\infty$. We determine the singularities of the lift and compute its Fourier expansion. It turns out that the lift is continuous but not differentiable along certain geodesics in the upper half-plane corresponding to the non-holomorphic principal part of the input. As an application, we obtain a generalization to higher level of the weight $2$ modular integral of Duke, Imamoglu and T\'oth. Further, we construct automorphic products associated to harmonic Maass forms.