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Generalized modular forms with a cuspidal divisor

Volume 197 / 2021

Quentin Gazda Acta Arithmetica 197 (2021), 247-257 MSC: Primary 11F03; Secondary 11F20, 11F12. DOI: 10.4064/aa190903-15-6 Published online: 23 October 2020

Abstract

In 2005, Kohnen proved that if $\Gamma =\Gamma _0(N)$ where $N$ is a square-free integer, then any modular function of weight $0$ for $\Gamma $ having a divisor supported at the cusps is an $\eta $-quotient. Under the assumption of rational Fourier coefficients, we are able to extend Kohnen’s result to the case where $N$ is the square of a prime. If the rationality condition does not hold, we show that the statement is no longer true by providing a family of counter-examples that generalizes naturally the Dedekind eta function. This paper fits within the framework of generalized modular forms in the sense of Knopp and Mason.

Authors

  • Quentin GazdaInstitut Camille Jordan
    Univ Lyon
    and
    Université Jean Monnet Saint-Étienne
    CNRS UMR 5208
    23 rue du Dr Paul Michelon
    42023 Saint-Étienne, France
    e-mail

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