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 X-0(pq) to compare the rational torsion subgroup of the Jacobian J(0)(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.