The eisenstein ideal and jacquet-langlands isogeny over function fields

Let p and q be two distinct prime ideals of F_q[T]. We use the Eisenstein ideal of the Hecke algebra of the Drinfeld modular curve X₀(pq) to compare the rational torsion subgroup of the Jacobian J0(pq) 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 Q.