{
  "id": "1808.09433",
  "version": "v1",
  "published": "2018-08-28T17:47:48.000Z",
  "updated": "2018-08-28T17:47:48.000Z",
  "title": "The Breuil--Mézard conjecture for function fields",
  "authors": [
    "Zijian Yao"
  ],
  "categories": [
    "math.NT",
    "math.RT"
  ],
  "abstract": "Let $K$ be a local function field of characteristic $l$, $\\mathbb{F}$ be a finite field over $\\mathbb{F}_p$ where $l \\ne p$, and $\\overline{\\rho}: G_K \\rightarrow \\text{GL}_n (\\mathbb{F})$ be a continuous representation. We apply the Taylor-Wiles-Kisin method over certain global function fields to construct a mod $p$ cycle map $\\overline{\\text{cyc}}$, from mod $p$ representations of $\\text{GL}_n (\\mathcal{O}_K)$ to the mod $p$ fibers of the framed universal deformation ring $R_{\\overline{\\rho}}^\\square$. This allows us to obtain a function field analog of the Breuil--M\\'ezard conjecture. Then we use the technique of close fields to show that our result is compatible with the Breuil-M\\'ezard conjecture for local number fields in the case of $l \\ne p$, obtained by Shotton.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-08-28T17:47:48.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "breuil-mézard conjecture",
      "breuil-mezard conjecture",
      "function field analog",
      "local function field",
      "local number fields"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}