{
  "id": "2106.04551",
  "version": "v1",
  "published": "2021-06-08T17:40:15.000Z",
  "updated": "2021-06-08T17:40:15.000Z",
  "title": "The Eisenstein ideal of weight $k$ and ranks of Hecke algebras",
  "authors": [
    "Shaunak V. Deo"
  ],
  "comment": "33 pages, Comments are welcome",
  "categories": [
    "math.NT"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-08T17:40:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F80",
      "11F25",
      "11F33"
    ],
    "keywords": [
      "hecke algebra",
      "global galois cohomology classes",
      "cuspidal modular forms",
      "results proven",
      "sufficient condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}