Zeta-polynomials, Hilbert polynomials, and the Eichler–Shimura identities
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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.
KeywordsPeriod polynomials Modular forms Zeta-polynomials Eichler–Shimura relations Hilbert polynomials
Mathematics Subject Classification11F11 11F67
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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