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

## TL;DR

This paper extends the concept of zeta-polynomials associated with modular forms by exploring analogous polynomials derived via the Rodriguez-Villegas transform, connecting them to Hilbert polynomials and Eichler-Shimura identities.

## Contribution

It introduces a broader class of polynomials related to modular forms, generalizing previous zeta-polynomials and establishing new connections with classical identities.

## Key findings

- Defined new polynomials using Rodriguez-Villegas transform
- Established functional equations similar to zeta-polynomials
- Linked these polynomials to Hilbert polynomials and 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.