Regularized inner products and errors of modularity

TL;DR

This paper introduces a new regularization for Petersson inner products of weakly holomorphic modular forms, enabling explicit evaluation and linking to harmonic Maass forms and cohomology classes.

Contribution

It generalizes existing regularizations, extends prior work on inner products, and connects modularity errors to cohomology in a broad, explicit framework.

Findings

Regularized inner products expressed via harmonic Maass form coefficients

Errors of modularity induce cocycles in first parabolic cohomology

Provides explicit representatives of cohomology classes

Abstract

We develop a regularization for Petersson inner products of arbitrary weakly holomorphic modular forms, generalizing several known regularizations. As one application, we extend work of Duke, Imamoglu, and Toth on regularized inner products of weakly holomorphic modular forms of weights $0$ and $3/2$. These regularized inner products can be evaluated in terms of the coefficients of holomorphic parts of harmonic Maass forms of dual weights. Moreover, we study the errors of modularity of the holomorphic parts of such a harmonic Maass forms and show that they induce cocyles in the first parabolic cohomology group introduced by Bruggeman, Choie, and the second author. This provides explicit representatives of the cohomology classes constructed abstractly and in a very general setting in their work.