On Eisenstein ideals and the cuspidal group of $J_0(N)$

摘要

Let $\mathcal{C}_N$ be the cuspidal subgroup of the Jacobian $J_0(N)$ for asquare-free integer $N>6$. For any Eisenstein maximal ideal $\mathfrak{m}$ ofthe Hecke ring of level $N$, we show that $\mathcal{C}_N[\mathfrak{m}]\neq 0$.To prove this, we calculate the index of an Eisenstein ideal $\mathcal{I}$contained in $\mathfrak{m}$ by computing the order of a cuspidal divisorannihilated by $\mathcal{I}$.