A magnetic modular form

TL;DR

This paper proves a divisibility conjecture related to Fourier coefficients of a meromorphic modular form, utilizing advanced lifts and operators, and constructs a family of such forms with this property.

Contribution

It introduces a proof of a conjecture on divisibility of Fourier coefficients and constructs a new family of meromorphic modular forms with this property.

Findings

Proved a conjecture on divisibility of Fourier coefficients.

Constructed a family of meromorphic modular forms with the divisibility property.

Utilized Borcherds' generalization of the Shimura lift and Hecke operators.

Abstract

In this paper, we prove a conjecture of Broadhurst and Zudilin \cite{BZ17} concerning a divisibility property of the Fourier coefficients of a meromorphic modular form using the generalization of the Shimura lift by Borcherds \cite{Borcherds98} and Hecke operators on vector-valued modular forms developed by Bruinier and Stein \cite{BS10}. Furthermore, we construct a family of meromorphic modular forms with this property.