On a theta lift related to the Shintani lift

TL;DR

This paper investigates a specific theta lift connecting harmonic weak Maass forms and classical modular forms, providing explicit Fourier expansions, criteria for non-vanishing L-values, and identities between cycle integrals.

Contribution

It introduces a new theta lift related to the Shintani lift, computes its Fourier expansion, and establishes criteria linking harmonic Maass forms to L-value non-vanishing.

Findings

Fourier expansion involves twisted traces and cycle integrals.

Provides a criterion for non-vanishing of central L-values.

Derives identities between cycle integrals of modular forms.

Abstract

We study a certain theta lift which maps weight $-2k$ to weight $1/2-k$ harmonic weak Maass forms for $k \in \mathbb{Z}, k \geq 0$, and which is closely related to the classical Shintani lift from weight $2k+2$ to weight $k+3/2$ cusp forms. We compute the Fourier expansion of the theta lift and show that it involves twisted traces of CM values and geodesic cycle integrals of the input function. As an application, we obtain a criterion for the non-vanishing of the central $L$-value of an integral weight newform $G$ in terms of the holomorphicity of the theta lift of a certain harmonic weak Maass form associated to $G$. Moreover, we derive interesting identities between cycle integrals of different kinds of modular forms.