Borcherds lifts of harmonic Maass forms and modular integrals

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 Tóth. Further, we construct automorphic products associated to harmonic Maass forms.

This work is a shortened version of a chapter of my 2018 PhD thesis. 26 pages. Comments welcome!

References in corpus (2)

• Inequalities for modified Bessel functions and their integrals
• Heegner divisors, $L$-functions and harmonic weak Maass forms