On the Torsion of the Mordell-Weil Group of the Jacobian of Drinfeld Modular Curves

Let Y0(p) be the Drinfeld modular curve parameterizing Drinfeld modules of rank two over Fq[T ] of general characteristic with Hecke level p-structure, where p ⊳ Fq[T ] is a prime ideal of degree d. Let J0(p) denote the Jacobian of the unique smooth irreducible projective curve containing Y0(p). Define N(p) = q−1 q−1 , if d is odd, and define N(p) = q −1 q2−1 , otherwise. We prove that the torsion subgroup of the group of Fq(T )-valued points of the abelian variety J0(p) is the cuspidal divisor group and has order N(p). Similarly the maximal μ-type finite étale subgroup-scheme of the abelian variety J0(p) is the Shimura group scheme and has order N(p). We reach our results through a study of the Eisenstein ideal E(p) of the Hecke algebra T(p) of the curve Y0(p). Along the way we prove that the completion of the Hecke algebra T(p) at any maximal ideal in the support of E(p) is Gorenstein. 2000 Mathematics Subject Classification: Primary 11G18; Secondary 11G09.