On the torsion of the Mordell-Weil group of the Jacobian of Drinfeld modular curves

Abstract

Let Y_0(\mathfrak p) be the Drinfeld modular curve parameterizing Drinfeld modules of rank two over \Bbb F_q[T] of general characteristic with Hecke level \mathfrak p -structure, where \mathfrak p\triangleleft\Bbb F_q[T] is a prime ideal of degree d . Let J_0(\mathfrak p) denote the Jacobian of the unique smooth irreducible projective curve containing Y_0(\mathfrak p) . Define N(\mathfrak p)=\frac{q^d-1}{q-1} , if d is odd, and define N(\mathfrak p)=\frac{q^d-1}{q^2-1} , otherwise. We prove that the torsion subgroup of the group of \Bbb F_q(T) -valued points of the abelian variety J_0(\mathfrak p) is the cuspidal divisor group and has order N(\mathfrak p) . Similarly the maximal \mu -type finite étale subgroup-scheme of the abelian variety J_0(\mathfrak p) is the Shimura group scheme and has order N(\mathfrak p) . We reach our results through a study of the Eisenstein ideal \mathfrak E(\mathfrak p) of the Hecke algebra \Bbb T(\mathfrak p) of the curve Y_0(\mathfrak p) . Along the way we prove that the completion of the Hecke algebra \Bbb T(\mathfrak p) at any maximal ideal in the support of \mathfrak E(\mathfrak p) is Gorenstein.