The rational torsion subgroups of Drinfeld modular Jacobians and Eisenstein pseudo-harmonic cochains

TL;DR

This paper investigates the structure of rational torsion subgroups of Jacobians of Drinfeld modular curves, establishing their relation to cuspidal divisor class groups and detailing their structure for primes not dividing specific quantities.

Contribution

It proves the equality of the -primary parts of the rational torsion subgroup and the cuspidal divisor class group for primes not dividing q(q-1), and characterizes the latter's structure.

Findings

-primary parts coincide for primes not dividing q(q-1)

Structure of cuspidal divisor class group determined for primes not dividing q-1

Provides explicit descriptions of torsion subgroups in Drinfeld modular Jacobians

Abstract

Let $\frak{n}$ be a square-free ideal of $\mathbb{F}_q[T]$. We study the rational torsion subgroup of the Jacobian variety $J_0(\frak{n})$ of the Drinfeld modular curve $X_0(\frak{n})$. We prove that for any prime number $\ell$ not dividing $q(q-1)$, the $\ell$-primary part of this group coincides with that of the cuspidal divisor class group. We further determine the structure of the $\ell$-primary part of the cuspidal divisor class group for any prime $\ell$ not dividing $q-1$.