The Eisenstein ideal with squarefree level

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 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. (C) 2021 Elsevier Inc. All rights reserved.