We use deformation theory of pseudorepresentations to study the analogue of Mazur's Eisenstein ideal with squarefree level. Given a prime number $p>3$ and a squarefree number N satisfying certain conditions, we study the Eisenstein part of the p-adic Hecke algebra for ${\mathrm{\Gamma }}_0(N)$, and show that it is a local complete intersection and isomorphic to a pseudodeformation ring. We also show that, in certain cases, the Eisenstein ideal is not principal and that the cuspidal quotient of the Hecke algebra is not Gorenstein. As a corollary, we prove that “multiplicity one” fails for the modular Jacobian $J_0(N)$ in these cases. In a particular case, this proves a conjecture of Ribet.

我们使用伪表示的变形理论来研究Mazur的Eisenstein理想与无平方水平的类似物。给定质数$p>3$以及满足某些条件的无平方数N，我们研究p -adic Hecke代数的Eisenstein部分${\mathrm{\Gamma }}_0（ñ）$，并证明它是局部完整的相交点，与伪变形环同构。我们还表明，在某些情况下，爱森斯坦理想主义不是主要的，赫克代数的尖峰商不是戈伦斯坦。因此，我们证明了模块化雅可比矩阵的“多重性”失败${Ĵ}_0（ñ）$在这些情况下。在特定情况下，这证明了里贝特的猜想。