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On cyclotomic factors of polynomials related to modular forms

MPS-Authors
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Heim,  Bernhard
Max Planck Institute for Mathematics, Max Planck Society;

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Luca,  Florian
Max Planck Institute for Mathematics, Max Planck Society;

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Neuhauser,  Markus
Max Planck Institute for Mathematics, Max Planck Society;

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https://doi.org/10.1007/s11139-017-9980-8
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Citation

Heim, B., Luca, F., & Neuhauser, M. (2019). On cyclotomic factors of polynomials related to modular forms. The Ramanujan Journal, 48(2), 445-458. doi:10.1007/s11139-017-9980-8.


Cite as: https://hdl.handle.net/21.11116/0000-0004-4246-3
Abstract
The Fourier coefficients of powers of the Dedekind eta function can be studied simultaneously. The vanishing of the coefficients varies from super lacunary (Euler, Jacobi identities) and lacunary (CM forms) to non-vanishing (Lehmer conjecture for the Ramanujan numbers). We study polynomials of degree n, whose roots control the vanishing of the nth Fourier coefficients of such powers. We prove that every root of unity appearing as any root of these polynomials has to be of order 2.