The divisibility of zeta functions of cyclotomic function fields

TL;DR

This paper extends Bernoulli-Goss polynomials and establishes a criterion for divisibility of zeta functions of cyclotomic function fields, demonstrating infinitely many such fields with zeta polynomials divisible by a given polynomial.

Contribution

It introduces a new divisibility criterion for zeta functions of cyclotomic function fields and proves the existence of infinitely many fields satisfying this criterion.

Findings

Established a divisibility criterion for zeta functions

Proved infinitely many cyclotomic function fields with divisible zeta polynomials exist

Generalized Bernoulli-Goss polynomials

Abstract

In this paper, we generalize Bernoulli-Goss polynomials, and give a criterion on the divisibility of zeta functions of cyclotomic function fields. As an application of our criterion, for a given polynomial $f(u)$, we prove that there are infinitely many cyclotomic function fields whose zeta polynomial is divided by $f(u)$.