On the Rational Cuspidal Divisor Class Groups of Drinfeld Modular Curves $X_0(\mathfrak{p}^r)$

Abstract

Let $\mathcal{C}(\mathfrak{p}^r)$ be the rational cuspidal divisor class group of the Drinfeld modular curve $X_0(\mathfrak{p}^r)$ for a prime power level $\mathfrak{p}^r\in \mathbb{F}_q[T]$. We relate the rational cuspidal divisors of degree $0$ on $X_0(\mathfrak{p}^r)$ with $Δ$-quotients, where $Δ$ is the Drinfeld discriminant function. As a result, we are able to determine explicitly the structure of $\mathcal{C}(\mathfrak{p}^r)$ for arbitrary prime $\mathfrak{p}\in \mathbb{F}_q[T]$ and $r\geq 2$.