The Eisenstein ideal with squarefree level

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 ${\Gamma }_0(N)$, 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.

49 pages, to appear in Adv. Math., revisions in response to referee report and some additions to the introduction

References in corpus (7)

• Pseudo-modularity and Iwasawa theory
• The rank of Mazur's Eisenstein ideal
• Ordinary pseudorepresentations and modular forms
• Deformation conditions for pseudorepresentations
• Companion forms and the structure of p-adic Hecke algebras
• On lifting and modularity of reducible residual Galois representations over imaginary quadratic fields
• The kernel of a rational Eisenstein prime at non-squarefree level

Cited by in corpus (6)

• Presentations of non-commutative deformation rings via $A_{\infty }$-algebras and applications to deformations of Galois representations and pseudorepresentations
• Iwasawa invariants in residually reducible Hida families
• A modular construction of unramified $p$-extensions of $\mathbb{Q}(N^{1/p})$
• Explicit non-Gorenstein R=T via rank bounds I: Deformation theory
• Relations among Ramanujan-Type Congruences II
• Higher congruences between newforms and Eisenstein series of squarefree level