Let $\frak{ p}$ and $\frak{ q}$ be two distinct prime ideals of $\mathbb{ F}_q[T]$. We use the Eisenstein ideal of the Hecke algebra of the Drinfeld modular curve $X_0(\frak{ p}\frak{ q})$ to compare the rational torsion subgroup of the Jacobian $J_0(\frak{ p}\frak{ q})$ with its subgroup generated by the cuspidal divisors, and to produce explicit examples of Jacquet-Langlands isogenies. Our results are stronger than what is currently known about the analogues of these problems over $\mathbb{ Q}$.
Categories: math.NT math.AG