Arithmetic properties of Fourier coefficients of meromorphic modular forms

Löbrich, Steffen; Schwagenscheidt, Markus

doi:10.2140/ant.2021.15.2381

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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

DOI: 10.2140/ant.2021.15.2381

Abstract

We investigate integrality and divisibility properties of Fourier coefficients of meromorphic modular forms of weight $2 k$ associated to positive definite integral binary quadratic forms. For example, we show that if there are no nontrivial cusp forms of weight $2 k$ , 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.

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

Vol. 15, No. 9, 2021