On rational Eisenstein primes and the rational cuspidal groups of modular Jacobian varieties

Let $N$ be a non-squarefree positive integer and let $\mathrm{\ell }$ be an odd prime such that ${\mathrm{\ell }}^2$ does not divide $N$. Consider the Hecke ring $\mathbb{T}(N)$ of weight $2$ for ${\mathrm{\Gamma }}_0(N)$, and its rational Eisenstein primes of $\mathbb{T}(N)$ containing $\mathrm{\ell }$, defined in Section 3. If $\mathfrak{m}$ is such a rational Eisenstein prime, then we prove that $\mathfrak{m}$ is of the form $(\mathrm{\ell },{\mathcal{I}}_{M,N}^D)$, where the ideal ${\mathcal{I}}_{M,N}^D$ of $\mathbb{T}(N)$ is also defined in Section 3. Furthermore, we prove that $\mathcal{C}(N)[\mathfrak{m}]\mathrm{\ne }0$, where $\mathcal{C}(N)$ is the rational cuspidal group of $J_0(N)$. To do this, we compute the precise order of the cuspidal divisor ${\mathcal{C}}_{M,N}^D$, defined in Section 4, and the index of ${\mathcal{I}}_{M,N}^D$ in $\mathbb{T}(N)\otimes {\mathbb{Z}}_{\mathrm{\ell }}$.