Polar harmonic Maass forms and their applications

TL;DR

This survey discusses recent developments in polar harmonic Maass forms, focusing on their properties, Fourier coefficients, and relations to Green's functions at CM-points, highlighting their significance in modular form theory.

Contribution

The paper compiles recent results on polar harmonic Maass forms, emphasizing new computational techniques and connections to Green's functions and CM-points.

Findings

Computed Fourier coefficients of meromorphic modular forms

Established relations between inner products and Green's functions

Connected polar harmonic Maass forms to classical modular objects

Abstract

In this survey, we present recent results of the authors about non-meromorphic modular objects known as polar harmonic Maass forms. These include the computation of Fourier coefficients of meromorphic modular forms and relations between inner products of meromorphic modular forms and higher Green's functions evaluated at CM-points.