{
  "id": "2211.03768",
  "version": "v1",
  "published": "2022-11-07T18:41:25.000Z",
  "updated": "2022-11-07T18:41:25.000Z",
  "title": "Lifting $G$-Valued Galois Representations when $\\ell \\neq p$",
  "authors": [
    "Jeremy Booher",
    "Sean Cotner",
    "Shiang Tang"
  ],
  "comment": "37 pages, comments welcome",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Let $\\ell$ and $p$ be distinct primes, $F$ an $\\ell$-adic field with absolute Galois group $\\Gamma_F$, and $k$ a finite field of characteristic $p$. For a split reductive group $G$, we investigate lifting continuous $\\overline{\\rho} : \\Gamma_F \\to G(k)$ to characteristic zero. We construct lifts of $\\overline{\\rho}$ and use the lift to construct a liftable deformation condition for $\\overline{\\rho}$ provided $p$ is large enough. This generalizes the minimally ramified deformation condition previously studied for classical groups. Doing so involves constructing ``decomposition types'' generalizing the notion of isotypic decomposition of a $\\textrm{GL}_n$-valued representation. This requires introducing and studying weakly reductive group schemes: smooth groups schemes with reductive identity component and a finite \\'{e}tale component group whose order is invertible on the base. Our work can be used to produce geometric lifts for global Galois representations, and we illustrate this for $G_2$-valued representations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-11-07T18:41:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F80"
    ],
    "keywords": [
      "valued galois representations",
      "weakly reductive group schemes",
      "valued representation",
      "produce geometric lifts",
      "smooth groups schemes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}