We study a certain theta lift which maps weight −2k to weight 1/2 − k harmonic weak Maass forms for k ∈ Z, k ≥ 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.
We derive finite rational formulas for the traces of cycle integrals of certain meromorphic modular forms. Moreover, we prove the modularity of a completion of the generating function of such traces. The theoretical framework for these results is an extension of the Shintani theta lift to meromorphic modular forms of positive even weight.
We define a regularized Shintani theta lift which maps weight 2k + 2 (k ∈ Z, k ≥ 0) harmonic Maass forms for congruence subgroups to (sesqui-)harmonic Maass forms of weight 3/2+k for the Weil representation of an even lattice of signature (1, 2). We show that its Fourier coefficients are given by traces of CM values and regularized cycle integrals of the input harmonic Maass form. Further, the Shintani theta lift is related via the ξ-operator to the Millson theta lift studied in our earlier work. We use this connection to construct ξ-preimages of Zagier's weight 1/2 generating series of singular moduli and of some of Ramanujan's mock theta functions. Contents 1. Introduction 1 2. Preliminaries 12 3. Regularized cycle integrals of harmonic Maass forms 15 4. Theta functions 20 5. The Shintani and the Millson theta lift 22 6. The Fourier expansion of the Shintani lift 26 References 43
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