The rank of Mazur's Eisenstein ideal

TL;DR

This paper investigates the structure and rank of Mazur's Eisenstein ideal using pseudodeformation theory, Galois cohomology, and Massey products, providing new proofs and generalizations of existing results.

Contribution

It introduces a novel approach to compute the rank of the Eisenstein part of the $p$-adic Hecke algebra using Massey products, extending previous work.

Findings

Computed the rank of the Hecke algebra in terms of Massey products.

Provided new proofs of Merel's and Mazur's results.

Generalized the understanding of the algebra's structure.

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.