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

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.

arXiv commentv2: 38 pages, made some changes and corrections following referee's comments, added a Corollary giving criteria for the principality of the Eisenstein ideal. Comments are welcome