Laurent expansions of meromorphic modular forms

TL;DR

This paper develops two methods to compute Laurent coefficients of meromorphic modular forms at CM points, one generalizing Rodriguez-Villegas and Zagier's approach and another using regularized theta lifts and harmonic Maass forms.

Contribution

It introduces two novel approaches for calculating Laurent coefficients of meromorphic modular forms at CM points, expanding existing computational techniques.

Findings

Laurent coefficients expressed as constant terms of recursive polynomials

Laurent coefficients related to Fourier coefficients of harmonic Maass forms

Methods applicable to a broad class of modular forms

Abstract

In this paper, we study the Laurent coefficients of meromorphic modular forms at CM 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.