On cyclotomic factors of polynomials related to modular forms
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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.
KeywordsDedekind eta function Fourier coefficients Integer-valued polynomials Cyclotomic polynomials
Mathematics Subject ClassificationPrimary 11D10 11F20 Secondary 11F30 11B83 11P84 11R18
We thank the anonymous referee for a careful reading of this manuscript and useful comments. This work was done while all three authors were visiting the Max Planck Institute for Mathematics in Bonn in July 2017. They thank this Institution for the invitation and excellent working conditions. In addition, B.H. and M.N. thank the RWTH Aachen, the Graduate school: Experimental and constructive algebra, chaired by G. Nebe, and the German University of Technology in Oman for their support.
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