Rational torsion of generalised Drinfeld modular Jacobians of prime power level

Abstract

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 $\ell$-primary part of the group $J_{0}(\mathfrak{n})_{\mathbf{m}}(\mathbb{F}_{q}(T))_{\rm{tor}}[\ell^{\infty}]$ is trivial for all primes $\ell$ not dividing $q(q^{2}-1)$. Our results establish a function field analogue to those of Yamazaki--Yang for the classical case.