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 ${\mathbb{F}}_q[T]$ of general characteristic with Hecke level $\mathfrak{p}$-structure, where $\mathfrak{p}◃{\mathbb{F}}_q[T]$ 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 ${\mathbb{F}}_q(T)$-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 $\mathbb{T}(\mathfrak{p})$ of the curve $Y_0(\mathfrak{p})$. Along the way we prove that the completion of the Hecke algebra $\mathbb{T}(\mathfrak{p})$ at any maximal ideal in the support of $\mathfrak{E}(\mathfrak{p})$ is Gorenstein.