Rational torsion of generalised Drinfeld modular Jacobians of prime power level

For a prime $\mathfrak{p}\subseteq {\mathbb{F}}_q[T]$ and a positive integer $r$, we consider the generalised Jacobian $J_0(\mathfrak{n}{)}_{\mathbf{m}}$ of the Drinfeld modular curve $X_0(\mathfrak{n})$ of level $\mathfrak{n}={\mathfrak{p}}^r$, with respect to the modulus~$\mathbf{m}$ consisting of all cusps on the modular curve. We show that the $\mathrm{\ell }$-primary part of the group $J_0(\mathfrak{n}{)}_{\mathbf{m}}({\mathbb{F}}_q(T){)}_{\mathrm{t}\mathrm{o}\mathrm{r}}[{\mathrm{\ell }}^{\mathrm{\infty }}]$ is trivial for all primes $\mathrm{\ell }$ not dividing $q(q^2-1)$. Our results establish a function field analogue to those of Yamazaki–Yang for the classical case.