The kernel of a rational Eisenstein prime at non-squarefree level

TL;DR

This paper precisely computes the dimension of the kernel associated with rational Eisenstein primes at non-squarefree levels in modular Jacobians, extending understanding in cases previously unresolved and proposing conjectures supported by computational evidence.

Contribution

It provides exact dimension formulas for kernels at non-squarefree levels and introduces conjectures for unresolved cases based on computational data.

Findings

Computed kernel dimensions under mild assumptions.

Proposed conjecture for complex cases based on Sage computations.

Extended the understanding of Eisenstein primes at non-squarefree levels.

Abstract

Let $\ell \geq 5$ be a prime and let $N$ be a non-squarefree integer not divisible by $\ell$. For a rational Eisenstein prime $\mathfrak{m}$ of the Hecke ring $\mathbb{T}(N)$ of level $N$ acting on $J_0(N)$, we precisely compute the dimension of the kernel $J_0(N)[\mathfrak{m}]$ under a mild assumption. In the case of level $qr^2$ which violates our mild assumption, we propose a conjecture based on Sage computations. Assuming this conjecture, we complete our computation in all the remaining cases.