On a theta lift related to the Shintani lift

Abstract

We study a certain theta lift which maps weight $-2k$ to weight $1/2-k$ harmonic weak Maass forms for $k\in \mathbb{Z},k\ge 0$, 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.

41 pages

References in corpus (2)

• Heegner divisors, $L$-functions and harmonic weak Maass forms
• Algebraic formulas for the coefficients of mock theta functions and Weyl vectors of Borcherds products

Cited by in corpus (6)

• Algebraic formulas for the coefficients of mock theta functions and Weyl vectors of Borcherds products
• Completions and algebraic formulas for the coefficients of Ramanujan's mock theta functions
• Traces of CM values and cycle integrals of polyharmonic Maass forms
• Traces of reciprocal singular moduli
• A modular framework for generalized Hurwitz class numbers III
• On p-adic haromonic Maass functions