Modularity of generating series of winding numbers

Abstract

The Shimura correspondence connects modular forms of integral weights and half-integral weights. One of the directions is realized by the Shintani lift, where the inputs are holomorphic differentials and the outputs are holomorphic modular forms of half-integral weight. In this article, we generalize this lift to differentials of the third kind. As an application, we obtain a modularity result concerning the generating series of winding numbers of closed geodesics on the modular curve.

Traces of reciprocal singular moduli

“…Remark Similar theta lifts of meromorphic modular forms were studied by Bruinier, Imamoglu, Funke and Li in [8] and by Bringmann and the authors of the present work in [1]. It was shown there that the generating series of traces of cycle integrals of meromorphic modular forms of positive even weight can be completed to real‐analytic modular forms of half‐integral weight whose images under the lowering operator are given by certain indefinite theta functions.…”

On the rationality of cycle integrals of meromorphic modular forms

“…The classical Shimura-Shintani correspondence establishes a Hecke equivariant isomorphism between the spaces of cusp forms of half-integral weight k + 1 2 and even integral weight 2k, with k ∈ N. Soon after its discovery by Shimura [18], this correspondence was realized by Shintani [19] as a theta lift, that is, as an integral constructed from a theta kernel in two variables. The classical Shintani theta lift for cusp forms was recently generalized to weakly holomorphic modular forms by Guerzhoy, Kane, and the second author [4], to harmonic Maass forms by the first and the third author [2], and to differentials of the third kind by Bruinier, Funke, Imamoglu, and Li [10]. Extending the results of [10], we also include meromorphic cusp forms of arbitrary positive even weight with poles of arbitrary order in the upper half-plane in the Shintani theta lift.…”

“…The classical Shintani theta lift for cusp forms was recently generalized to weakly holomorphic modular forms by Guerzhoy, Kane, and the second author [4], to harmonic Maass forms by the first and the third author [2], and to differentials of the third kind by Bruinier, Funke, Imamoglu, and Li [10]. Extending the results of [10], we also include meromorphic cusp forms of arbitrary positive even weight with poles of arbitrary order in the upper half-plane in the Shintani theta lift.…”

“…Every meromorphic modular form can be written as a sum of a meromorphic cusp form, a weakly holomorphic modular form, and, if k = 1, a multiple of j ′ /j, with j the usual modular j-invariant. Since the Shintani theta lifts of weakly holomorphic modular forms and of the meromorphic modular form j ′ /j have already been determined in [2,10], we restrict our attention to theta lifts of meromorphic cusp forms.…”

“…(2) Using a vector-valued setting as in [2,10], the methods of the present paper can be applied to compute the Shintani theta lift of meromorphic cusp forms for Γ 0 (N ).…”