The Eisenstein ideal with squarefree level

We use pseudodeformation theory to study the analogue of Mazur's Eisenstein ideal with certain squarefree levels. 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 \(\Gamma_0(N)\), 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 in these cases. In a particular case, this proves a conjecture of Ribet.