Abstract
Recently, Ono et al. answered problems of Manin by defining zeta-polynomials \(Z_f(s)\) for even weight newforms \(f\in S_k(\varGamma _0(N)\); these polynomials can be defined by applying the “Rodriguez-Villegas transform” to the period polynomial of f. It is known that these zeta-polynomials satisfy a functional equation \(Z_f(s) = \pm \, Z_f(1-s)\) and they have a conjectural arithmetic-geometric interpretation. Here, we give analogous results for a slightly larger class of polynomials which are also defined using the Rodriguez–Villegas transform.
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The author thanks Nick Andersen, Maddie Locus Dawsey, Michael Griffin, Tim Huber, Larry Rolen, and Armin Straub for their helpful discussions and correspondence.
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Jameson, M. Zeta-polynomials, Hilbert polynomials, and the Eichler–Shimura identities. Res Math Sci 6, 27 (2019). https://doi.org/10.1007/s40687-019-0190-4
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DOI: https://doi.org/10.1007/s40687-019-0190-4