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

Abstract

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