On the Fourier coefficients of meromorphic Jacobi forms

TL;DR

This paper explores the automorphic properties of Fourier coefficients of meromorphic Jacobi forms, showing they are related to almost harmonic Maass forms and detailing their completions based on Laurent coefficients and known real analytic functions.

Contribution

It extends previous work by characterizing Fourier coefficients of meromorphic Jacobi forms as parts of almost harmonic Maass forms and describes their unique completions.

Findings

Fourier coefficients are holomorphic parts of almost harmonic Maass forms

Completions are uniquely determined by Laurent coefficients and real analytic functions

Provides explicit descriptions of the automorphic properties of these coefficients

Abstract

In this paper, we describe the automorphic properties of the Fourier coefficients of meromorphic Jacobi forms. Extending results of Dabholkar, Murthy, and Zagier, and Bringmann and Folsom, we prove that the canonical Fourier coefficients of a meromorphic Jacobi form $\phi(z;\tau)$ are the holomorphic parts of some (vector-valued) almost harmonic Maass forms. We also give a precise description of their completions, which turn out to be uniquely determined by the Laurent coefficients of $\phi$ at each pole, as well as some well known real analytic functions, that appear for instance in the completion of Appell-Lerch sums.