#### Vol. 15, No. 9, 2021

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Arithmetic properties of Fourier coefficients of meromorphic modular forms

### Steffen Löbrich and Markus Schwagenscheidt

Vol. 15 (2021), No. 9, 2381–2401
##### 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 nontrivial 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 nonvanishing 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.

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##### Keywords
meromorphic modular forms, modular forms of half-integral weight, Fourier coefficients, integrality, divisibility, nonvanishing, sign changes
##### Mathematical Subject Classification
Primary: 11F30, 11F33, 11F37
Secondary: 11F25, 11F27
##### Milestones
Revised: 17 March 2021
Accepted: 22 March 2021
Published: 23 December 2021
##### Authors
 Steffen Löbrich Korteweg-de Vries Institute for Mathematics University of Amsterdam Netherlands Markus Schwagenscheidt Mathematics Department ETH Zürich Switzerland