Rational torsion of generalised Drinfeld modular Jacobians of prime power level

TL;DR

This paper investigates the rational torsion subgroup of generalized Jacobians of Drinfeld modular curves at prime power levels, showing it is trivial for almost all primes, thus extending classical results to the function field setting.

Contribution

It proves the triviality of the $ ext{l}$-primary torsion subgroup of these Jacobians over function fields for all primes not dividing $q(q^{2}-1)$, generalizing known classical results.

Findings

The $ ext{l}$-primary torsion subgroup is trivial for all primes $ ext{l}$ not dividing $q(q^{2}-1)$.

Establishes a function field analogue of Yamazaki--Yang's classical results.

Extends understanding of torsion in Drinfeld modular Jacobians at prime power levels.

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.