On the $p$-divisibility of class numbers of cyclotomic function fields

Abstract

Let $\mathbb{F}_q$ be the finite field with $q=p^r$ elements, where $p$ is a prime. For a monic $m \in \mathbb{F}_q[T]$, let $h_m$ be the class number of the $m$th cyclotomic function field. The goal of this paper is to determine the $p$-divisibility of $h_m$ when $q = p$ $(r \ge 2)$ and $\deg m=2$.