Arithmetic properties of Fourier coefficients of meromorphic modular forms

TL;DR

This paper explores the integrality, divisibility, and sign properties of Fourier coefficients of meromorphic modular forms linked to quadratic forms, revealing new divisibility patterns and relations to classical functions.

Contribution

It establishes divisibility results for Fourier coefficients when certain cusp forms are absent, and connects these coefficients to the j-function and partition function.

Findings

Fourier coefficients are divisible by n^{k-1} under specific conditions

Coefficients are non-vanishing and exhibit constant or alternating signs

Relations are derived between Fourier coefficients, the j-function, and the partition function

Abstract

We investigate integrality and divisibility properties of Fourier coefficients of meromorphic modular forms of weight $2k$ associated to positive definite integral binary quadratic forms. For example, we show that if there are no non-trivial cusp forms of weight $2k$, then the $n$-th coefficients of these meromorphic modular forms are divisible by $n^{k-1}$ for every natural number $n$. Moreover, we prove that their coefficients are non-vanishing and have either constant or alternating signs. Finally, we obtain a relation between the Fourier coefficients of meromorphic modular forms, the coefficients of the $j$-function, and the partition function.