On the rationality of cycle integrals of meromorphic modular forms

TL;DR

This paper develops rational formulas for cycle integral traces of meromorphic modular forms and proves the modularity of their generating functions, extending the Shintani theta lift to meromorphic cases.

Contribution

It introduces finite rational formulas for cycle integral traces and extends the Shintani theta lift to meromorphic modular forms of positive even weight.

Findings

Derived rational formulas for cycle integral traces.

Proved modularity of the generating function of these traces.

Extended the Shintani theta lift to meromorphic modular forms.

Abstract

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.