The rational torsion subgroup of \(J_0(\mathfrak {p}^r)\)

Let \(\mathfrak {n}=\mathfrak {p}^r\) $\mathfrak{n}={\mathfrak{p}}^r$ be a prime power ideal of \(\mathbb {F}_q[T]\) ${\mathbb{F}}_q[T]$ with \(r\ge 2\) $r\ge 2$ . We study the rational torsion subgroup \(\mathcal {T}(\mathfrak {p}^r)\) $\mathcal{T}({\mathfrak{p}}^r)$ of the Drinfeld modular Jacobian \(J_0(\mathfrak {p}^r)\) $J_0({\mathfrak{p}}^r)$ . We prove that the prime-to- \(q(q-1)\) $q(q-1)$ part of \(\mathcal {T}(\mathfrak {p}^r)\) $\mathcal{T}({\mathfrak{p}}^r)$ is equal to that of the rational cuspidal divisor class group \(\mathcal {C}(\mathfrak {p}^r)\) $\mathcal{C}({\mathfrak{p}}^r)$ of the Drinfeld modular curve \(X_0(\mathfrak {p}^r)\) $X_0({\mathfrak{p}}^r)$ . As we completely computed the structure of \(\mathcal {C}(\mathfrak {p}^r)\) $\mathcal{C}({\mathfrak{p}}^r)$ , it also determines the structure of the prime-to- \(q(q-1)\) $q(q-1)$ part of \(\mathcal {T}(\mathfrak {p}^r)\) $\mathcal{T}({\mathfrak{p}}^r)$ .