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

TL;DR

This paper characterizes rational Eisenstein primes in the Hecke ring for non-squarefree levels and demonstrates their connection to the rational cuspidal group of modular Jacobians, providing explicit computations of related divisors and ideals.

Contribution

It explicitly describes the structure of rational Eisenstein primes for certain levels and proves their non-trivial action on the rational cuspidal group, with detailed divisor and ideal index calculations.

Findings

Rational Eisenstein primes are of the form $( ext{prime}, ext{ideal})$

The rational cuspidal group has non-zero elements fixed by these primes

Explicit orders of cuspidal divisors and ideal indices are computed

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 $(\ell, ~\mathcal{I}^D_{M, N})$, where the ideal $\mathcal{I}^D_{M, N}$ of $\mathbb{T}(N)$ is also defined in Section 3. Furthermore, we prove that $\mathcal{C}(N)[\mathfrak{m}] \neq 0$, 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 $\mathcal{C}^D_{M, N}$, defined in Section 4, and the index of $\mathcal{I}^D_{M, N}$ in $\mathbb{T}(N)\otimes \mathbb{Z}_\ell$.