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

Abstract

Let \(\mathfrak n\) be a square-free ideal of \(\mathbb {F}_q[T]\). We study the rational torsion subgroup of the Jacobian variety \(J_0(\mathfrak n)\) of the Drinfeld modular curve \(X_0(\mathfrak n)\). 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 \(q-1\).