Regularized inner products and weakly holomorphic Hecke eigenforms

TL;DR

This paper reveals that repeated differentiation of weak cusp forms characterizes a specific orthogonal subspace related to weakly holomorphic modular forms, offering a new perspective on weakly holomorphic Hecke eigenforms.

Contribution

It introduces a novel interpretation of weakly holomorphic Hecke eigenforms through the orthogonal complement of differentiated weak cusp forms.

Findings

Differentiation maps weak cusp forms to an orthogonal subspace.

Provides a new understanding of weakly holomorphic Hecke eigenforms.

Establishes a link between differentiation and orthogonality in modular forms.

Abstract

We show that the image of repeated differentiation on weak cusp forms is precisely the subspace which is orthogonal to the space of weakly holomorphic modular forms. This gives a new interpretation of the weakly holomorphic Hecke eigenforms.