{
  "id": "1910.03205",
  "version": "v1",
  "published": "2019-10-08T04:16:49.000Z",
  "updated": "2019-10-08T04:16:49.000Z",
  "title": "On a conjecture of Sharifi and Mazur's Eisenstein ideal",
  "authors": [
    "Emmanuel Lecouturier",
    "Jun Wang"
  ],
  "comment": "22 pages. Comments welcome!",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $N$ and $p$ be prime numbers $\\geq 5$ such that $p$ divides $N-1$. Let $I$ be Mazur's Eisenstein ideal of level $N$ and $H_+$ be the plus part of $H_1(X_0(N), \\mathbf{Z}_p)$ for the complex conjugation. We give a conjectural explicit description of the group $I\\cdot H_+/I^2\\cdot H_+$ in terms of the second $K$-group of the cyclotomic field $\\mathbf{Q}(\\zeta_N)$. We prove that this conjecture follows from a conjecture of Sharifi about some Eisenstein ideal of level $\\Gamma_1(N)$. Following the work of Fukaya--Kato, we prove partial results on Sharifi's conjecture. This allows us to prove partial results on our conjecture.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-08T04:16:49.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "mazurs eisenstein ideal",
      "partial results",
      "conjectural explicit description",
      "sharifis conjecture",
      "cyclotomic field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}