Zagier duality and integrality for Fourier coefficients for weakly holomorphic modular forms

TL;DR

This paper extends the understanding of Fourier coefficients in weakly holomorphic modular forms by generalizing isomorphisms beyond real quadratic fields, proving Zagier duality, and exploring integrality questions for these coefficients.

Contribution

It generalizes isomorphisms for discriminant forms, proves Zagier duality for canonical bases, and raises questions about the integrality of Fourier coefficients.

Findings

Established generalized isomorphisms for broader discriminant forms

Proved Zagier duality for canonical bases of modular forms

Highlighted open questions on the integrality of Fourier coefficients

Abstract

In this note, we generalize the isomorphisms to the case when the discriminant form is not necessarily induced from real quadratic fields. In particular, this general setting includes all the subspaces with epsilon-conditions, only two spacial cases of which were treated before. With this established, we shall prove the Zagier duality for canonical bases. Finally, we raise a question on the integrality of the Fourier coefficients of these bases elements, or equivalently we concern the existence of a Miller-like basis for vector valued modular forms.