The Eisenstein ideal of weight $k$ and ranks of Hecke algebras

TL;DR

This paper investigates the structure of Hecke algebras acting on modular forms of weight $k$ at Eisenstein ideals, providing criteria for their ranks and establishing $R=\mathbb{T}$ theorems using deformation theory.

Contribution

It introduces a new deformation-theoretic approach to analyze the Eisenstein ideal and ranks of Hecke algebras, extending previous results to higher weights and establishing new $R=\mathbb{T}$ theorems.

Findings

Criteria for the $\mathbb{Z}_p$-rank of Hecke algebras based on Galois cohomology

Recovery of results for weight $k=2$ using new methods

Proof of $R=\mathbb{T}$ theorems under specific hypotheses

Abstract

Let $p$ and $\ell$ be primes such that $p > 3$ and $p \mid \ell-1$ and $k$ be an even integer. We use deformation theory of pseudo-representations to study the completion of the Hecke algebra acting on the space of cuspidal modular forms of weight $k$ and level $\Gamma_0(\ell)$ at the maximal Eisenstein ideal containing $p$. We give a necessary and sufficient condition for the $\mathbb{Z}_p$-rank of this Hecke algebra to be greater than $1$ in terms of vanishing of the cup products of certain global Galois cohomology classes. We also recover some of the results proven by Wake and Wang-Erickson for $k=2$ using our methods. In addition, we prove some $R=\mathbb{T}$ theorems under certain hypothesis.