Laurent expansions of meromorphic modular forms

Abstract

In this paper, we study the Laurent coefficients of meromorphic modular forms at complex multiplication points by giving two approaches of computing them. The first is a generalization of the method of Rodriguez-Villegas and Zagier, which expresses the Laurent coefficients as constant terms of a family of polynomials obtained through recursion. The second applies to meromorphic modular forms that are regularized theta lifts, and expresses their Laurent coefficients in terms of Fourier coefficients of harmonic Maass forms.