The rank of Mazur's Eisenstein ideal

Abstract

We use pseudodeformation theory to study Mazur's Eisenstein ideal. Given prime numbers $N$ and $p>3$, we study the Eisenstein part of the $p$-adic Hecke algebra for ${\Gamma }_0(N)$. We compute the rank of this Hecke algebra (and, more generally, its Newton polygon) in terms of Massey products in Galois cohomology, answering a question of Mazur and generalizing a result of Calegari-Emerton. We also also give new proofs of Merel's result on this rank and of Mazur's results on the structure of the Hecke algebra.

63 pages. Final version. Improvements to exposition and minor corrections, added dedication. To appear in Duke Math J

References in corpus (6)

• Derived Galois deformation rings
• Algebraic families of Galois representations and potentially semi-stable pseudodeformation rings
• Sur le p-rang du groupe des classes de Q(N^1/p)
• Pseudo-modularity and Iwasawa theory
• Deformation conditions for pseudorepresentations
• Class groups of Kummer extensions via cup products in Galois cohomology

Cited by in corpus (5)

• The Eisenstein ideal with squarefree level
• Explicit non-Gorenstein R=T via rank bounds II: Computational aspects
• Iwasawa invariants in residually reducible Hida families
• Explicit non-Gorenstein R=T via rank bounds I: Deformation theory
• Class group statistics for torsion fields generated by elliptic curves