Meromorphic Modular Forms with Rational Cycle Integrals

Steffen Löbrich,Markus Schwagenscheidt

doi:10.1093/imrn/rnaa104

Abstract

AbstractWe study rationality properties of geodesic cycle integrals of meromorphic modular forms associated to positive definite binary quadratic forms. In particular, we obtain finite rational formulas for the cycle integrals of suitable linear combinations of these meromorphic modular forms.

Highlights

One of the fundamental results in the classical theory of modular forms is the fact that the vector spaces of modular forms are spanned by forms with rational Fourier coefficients
There are other natural rational structures on these spaces, for example coming from the rationality of periods or cycle integrals of modular forms
This was first shown by Kohnen and Zagier in [20], where they proved the rationality of the even periods of the cusp forms fk,D(z)

Summary

Introduction

One of the fundamental results in the classical theory of modular forms is the fact that the vector spaces of modular forms are spanned by forms with rational Fourier coefficients. There are other natural rational structures on these spaces, for example coming from the rationality of periods or cycle integrals of modular forms. This was first shown by Kohnen and Zagier in [20], where they proved the rationality of the even periods of the cusp forms (1.1). These forms recently attracted some attention, starting with the work of Bengoechea [2] on the rationality properties of their Fourier coefficients Their regularized inner products and connections to locally harmonic Maass forms were investigated by Bringmann, Kane, and von Pippich [8] and the first author [21]. We remark that our results generalize the rationality results of [1] in several aspects, using a very different proof

Statement of results

Preliminaries

Locally harmonic Maass forms