Regularized inner products and meromorphic modular forms

Abstract

In this talk, we consider a regularization of Petersson's inner product which is well-defined (and finite) between two meromorphic modular forms and agrees with Petersson's inner product whenever the latter exists. We take the inner product between a special family of meromorphic modular forms and show a connection with the automorphic Green's function and certain functions which are called polar harmonic Maass forms. We then discuss some applications of these polar harmonic Maass forms, including formulas and asymptotics for Fourier coefficients of meromorphic modular forms, construction of a basis of meromorphic modular forms of non-positive weight, and an algorithm to compute the divisor of a given meromorphic modular form given only its Fourier expansion. Most of the talk is joint work with Kathrin Bringmann and Anna von Pippich, while the first two applications are joint work with Kathrin Bringmann and the application to divisors of modular forms is joint work with Kathrin Bringmann, Steffen Loebrich, Ken Ono, and Larry Rolen.