We study the Eisenstein ideal of Drinfeld modular curves of small levels, and the relation of the Eisenstein ideal to the cuspidal divisor group and the component groups of Jacobians of Drinfeld modular curves. We prove that the characteristic of the function field is an Eisenstein prime number when the level is an arbitrary non square-free ideal of F q [T ] not equal to a square of a prime.

Abstract. Let n be a square-free ideal of F q [T ]. We study the rational torsion subgroup of the Jacobian variety J 0 (n) of the Drinfeld modular curve X 0 (n). We prove that for any prime number ℓ not dividing q(q − 1), the ℓ-primary part of this group coincides with that of the cuspidal divisor class group. We further determine the structure of the ℓ-primary part of the cuspidal divisor class group for any prime ℓ not dividing q − 1.

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.

We consider the generalized Jacobian J of the modular curve X 0 (N ) of level N with respect to a reduced divisor consisting of all cusps. Supposing N is square free, we explicitly determine the structure of the Q-rational torsion points on J up to 6-primary torsion. The result depicts a fuller picture than [18] where the case of prime power level was studied. We also obtain an analogous result for Drinfeld modular curves. Our proof relies on similar results for classical Jacobians due to Ohta, Papikian and the first author. We also discuss the Hecke action on J and its Eisenstein property.