Zeta-polynomials, Hilbert polynomials, and the Eichler–Shimura identities

  • Marie JamesonEmail author


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.


Period polynomials Modular forms Zeta-polynomials Eichler–Shimura relations Hilbert polynomials 

Mathematics Subject Classification

11F11 11F67 


Author's contributions


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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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of TennesseeKnoxvilleUSA

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