Cycle integrals of a sesqui-harmonic Maass form of weight zero

TL;DR

This paper establishes a connection between Fourier coefficients of weight 3/2 mock modular forms and cycle integrals of a sesqui-harmonic Maass form of weight zero, extending the understanding of traces of singular moduli.

Contribution

It introduces a new relationship between Fourier coefficients of certain mock modular forms and cycle integrals of sesqui-harmonic Maass forms, expanding the theory of modular forms and their cycle integrals.

Findings

Fourier coefficients are sums of cycle integrals of a sesqui-harmonic Maass form.

These sums can be expressed as regularized inner products of weight 1/2 modular forms.

The work generalizes the connection between mock modular forms and traces of singular moduli.

Abstract

Borcherds-Zagier bases of the spaces of weakly holomorphic modular forms of weights 1/2 and 3/2 share the Fourier coefficients which are traces of singular moduli. Recently, Duke, Imamo\={g}lu, and T\'{o}th have constructed a basis of the space of weight 1/2 mock modular forms, each member in which has Zagier's generating series of traces of singular moduli as its shadow. They also showed that Fourier coefficients of their mock modular forms are sums of cycle integrals of the $j$-function which are real quadratic analogues of singular moduli. In this paper, we prove the Fourier coefficients of a basis of the space of weight 3/2 mock modular forms are sums of cycle integrals of a sesqui-harmonic Maass form of weight zero whose image under hyperbolic Laplacian is the $j$-function. Furthermore, we express these sums as regularized inner products of weakly holomorphic modular forms of weight 1/2.