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 nk1 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
Received: 20 October 2020
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