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Arithmetic properties
of Fourier coefficients of meromorphic modular forms
Steffen Löbrich and Markus Schwagenscheidt |
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Vol. 15 (2021), No. 9, 2381–2401
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Abstract |
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We investigate integrality and divisibility properties of Fourier coefficients of meromorphic modular forms of weight associated to positive definite integral binary quadratic forms. For example, we show that if there are no nontrivial cusp forms of weight , then the -th coefficients of these meromorphic modular forms are divisible by for every natural number . 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 -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
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Mathematical Subject Classification
Primary: 11F30, 11F33, 11F37
Secondary: 11F25, 11F27
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Milestones
Received: 20 October 2020
Revised: 17 March 2021
Accepted: 22 March 2021
Published: 23 December 2021
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Authors |
| Steffen Löbrich | |
| Korteweg-de Vries Institute for
Mathematics University of Amsterdam Netherlands |
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| Markus Schwagenscheidt | |
| Mathematics Department ETH Zürich Switzerland |
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