{
  "id": "1608.01784",
  "version": "v1",
  "published": "2016-08-05T07:17:53.000Z",
  "updated": "2016-08-05T07:17:53.000Z",
  "title": "The Breuil--Mézard conjecture when $l \\neq p$",
  "authors": [
    "Jack Shotton"
  ],
  "comment": "51 pages. Submitted",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $l$ and $p$ be primes, let $F/\\mathbb{Q}_p$ be a finite extension with absolute Galois group $G_F$, let $\\mathbb{F}$ be a finite field of characteristic $l$, and let $\\bar{\\rho} : G_F \\rightarrow GL_n(\\mathbb{F})$ be a continuous representation. Let $R^\\square(\\bar{\\rho})$ be the universal framed deformation ring for $\\bar{\\rho}$. If $l = p$, then the Breuil--M\\'{e}zard conjecture (as formulated by Emerton and Gee) relates the mod $l$ reduction of certain cycles in $R^\\square(\\bar{\\rho})$ to the mod $l$ reduction of certain representations of $GL_n(\\mathcal{O}_F)$. We state an analogue of the Breuil--M\\'{e}zard conjecture when $l \\neq p$, and prove it whenever $l > 2$ using automorphy lifting theorems. We give a local proof when $l$ is \"quasi-banal\" for $F$ and $\\bar{\\rho}$ is tamely ramified. We also analyse the reduction modulo $l$ of the types $\\sigma(\\tau)$ defined by Schneider and Zink.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-08-05T07:17:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "breuil-mézard conjecture",
      "absolute galois group",
      "reduction modulo",
      "finite field",
      "finite extension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 51,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}