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

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 $\mathrm{\Delta }$-quotients, where $\mathrm{\Delta }$ 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 $r\ge 2$.