{
  "id": "1812.04208",
  "version": "v1",
  "published": "2018-12-11T03:42:18.000Z",
  "updated": "2018-12-11T03:42:18.000Z",
  "title": "Change of coefficients in the $p \\neq \\ell$ local Langlands correspondence for $GL_n$",
  "authors": [
    "Tibor Backhausz"
  ],
  "comment": "23 pages; comments welcome",
  "categories": [
    "math.NT",
    "math.RT"
  ],
  "abstract": "Let $\\ell$ and $p$ be distinct primes, $n$ a positive integer, $F_\\ell$ an $\\ell$-adic local field of characteristic $0,$ and let $W(k)$ denote the ring of Witt vectors over an algebraically closed field of characteristic $p$. Work of Emerton-Helm, Helm and Helm-Moss defines and constructs a smooth $A[{GL}_n(F_\\ell)]$-module $\\tilde{\\pi}(\\rho_A)$ for a continuous Galois representation $\\rho_A : G_{F_\\ell} \\to {GL}_n(A)$ over a $p$-torsionfree reduced complete local $W(k)$-algebra $A$ interpolating the local Langlands correspondence. However, since $\\tilde{\\pi}$ is not a functor, there is no clear way to speak about the local Langlands correspondence over non-reduced or finite characteristic $W(k)$-algebras. We describe two natural and reasonable variants of the local Langlands correspondence with arbitrary complete local $W(k)$-algebras as coefficients. They are isomorphic when evaluated on the universal framed deformation of a Galois representation $\\overline{\\rho}$ over $k$, and more generally we find a surjection in one direction. In many cases, including $n=2$ or $3,$ they both recover $\\tilde{\\pi}(\\rho)$ when $\\rho$ has coefficients in a finite extension of $W(k)[p^{-1}].$ On the Galois side, this requires finding minimal lifts between Galois deformations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-12-11T03:42:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F70",
      "11F80"
    ],
    "keywords": [
      "local langlands correspondence",
      "coefficients",
      "galois representation",
      "torsionfree reduced complete local",
      "arbitrary complete local"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}