Zagier duality for level $p$ weakly holomorphic modular forms

TL;DR

This paper establishes Zagier duality for Fourier coefficients of weakly holomorphic modular forms at prime levels between 11 and 37, providing new duality results and generating functions for genus 1 levels.

Contribution

It proves Zagier duality for prime level weakly holomorphic modular forms and derives generating functions for genus 1 levels, extending previous duality results.

Findings

Proved Zagier duality for levels 11 to 37.

Derived generating functions for genus 1 levels.

Established duality for an infinite class of prime levels.

Abstract

We prove Zagier duality between the Fourier coefficients of canonical bases for spaces of weakly holomorphic modular forms of prime level $p$ with $11 \leq p \leq 37$ with poles only at the cusp at $\infty$, and special cases of duality for an infinite class of prime levels. We derive generating functions for the bases for genus 1 levels.