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

Abstract

In 2017, Ono, Rolen, and Sprung [ORS17] answered problems of Manin [Man16] by defining zeta-polynomials $Z_f(s)$ for even weight newforms $f\in S_k({\Gamma }_0(N)$; 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.

This is a pre-print. Comments are appreciated

References in corpus (2)

• The Riemann Hypothesis For Period Polynomials Of Modular Forms
• Local zeta factors and geometries under Spec Z