Arithmetic properties of Fourier coefficients of meromorphic modular forms

2020/10/13 by Steffen Löbrich, Markus Schwagenscheidt, Löbrich, Steffen +1 · 1 citation
Mathematics · #11F25 #11F27 #11F30 #11F33 #11F37 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2010.06297

openalex publication_date 2020/10/13 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28

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.

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• Elliptic curves and Fourier coefficients of meromorphic modular forms

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