Abstract: 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 Γ0(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.
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