On the torsion of the Mordell-Weil group of the Jacobian of Drinfeld modular curves
Let be the Drinfeld modular curve parameterizing Drinfeld modules of rank two over of general characteristic with Hecke level -structure, where is a prime ideal of degree . Let denote the Jacobian of the unique smooth irreducible projective curve containing . Define , if is odd, and define , otherwise. We prove that the torsion subgroup of the group of -valued points of the abelian variety is the cuspidal divisor group and has order . Similarly the maximal -type finite étale subgroup-scheme of the abelian variety is the Shimura group scheme and has order . We reach our results through a study of the Eisenstein ideal of the Hecke algebra of the curve . Along the way we prove that the completion of the Hecke algebra at any maximal ideal in the support of is Gorenstein.
Cite this article
Ambrus Pál, On the torsion of the Mordell-Weil group of the Jacobian of Drinfeld modular curves. Doc. Math. 10 (2005), pp. 131–198DOI 10.4171/DM/185