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

Abstract

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

arXiv commentMany arguments are clarified, and many details are filled in