ncts

Fix a non-zero ideal $\mathfrak{n}$ of $\mathbb{F}_q[T]$ . Let $\mathcal{T}(\mathfrak{n})$ be the rational torsion subgroup of the Drinfeld modular Jacobian $J_0(\mathfrak{n})$ . A generalized Ogg's conjecture states that $\mathcal{T}(\mathfrak{n})$ coincides with the rational cuspidal divisor class group $\mathcal{C}(\mathfrak{n})$ of the Drinfeld modular curve $X_0(\mathfrak{n})$ . Recently, we proved that for any prime-power ideal $\mathfrak{p}^r$ of $\mathbb{F}_q[T]$ , the prime-to- $q(q-1)$ part of $\mathcal{T}(\mathfrak{p}^r)$ is equal to that of $\mathcal{C}(\mathfrak{p}^r)$ by studying the Hecke operators and the Eisenstein ideal of level $\mathfrak{p}^r$ . Moreover, by relating the rational cuspidal divisors of degree $0$ on $X_0(\mathfrak{p}^r)$ with $\Delta$ -quotients, where $\Delta$ is the Drinfeld discriminant function, we are able to compute explicitly the structure of $\mathcal{C}(\mathfrak{p}^r)$ . As a result, the structure of the prime-to- $q(q-1)$ part of $\mathcal{T}(\mathfrak{p}^r)$ is completely determined.