{
  "id": "2306.02493",
  "version": "v1",
  "published": "2023-06-04T22:00:20.000Z",
  "updated": "2023-06-04T22:00:20.000Z",
  "title": "Galois representations with large image in global Langlands correspondence",
  "authors": [
    "Adrian Zenteno"
  ],
  "comment": "This article supersedes arXiv:2008.00556",
  "categories": [
    "math.NT",
    "math.GR"
  ],
  "abstract": "The global Langlands conjecture for $\\text{GL}_n$ over a number field $F$ predicts a correspondence between certain algebraic automorphic representations $\\pi$ of $\\text{GL}_n(\\mathbb{A}_F)$ and certain families $\\{ \\rho_{\\pi,\\ell} \\}_\\ell$ of $n$-dimensional $\\ell$-adic Galois representations of $\\text{Gal}(\\overline{F}/F)$. In general, it is expected that the image of the residual Galois representations $\\overline{\\rho}_{\\pi,\\ell}$ of the $\\rho_{\\pi,\\ell}$'s should be as large as possible for almost all primes $\\ell$, unless there is an automorphic reason for it does not happen. In this paper, we study the images of certain compatible systems of Galois representations $\\{\\rho_{\\pi,\\ell} \\}_\\ell$ associated to regular algebraic, polarizable, cuspidal automorphic representations $\\pi$ of $\\text{GL}_n(\\mathbb{A}_F)$ by using only standard techniques and currently available tools (e.g., Fontaine-Laffaille theory, Serre's modularity conjecture, classification of the maximal subgroups of Lie type groups, and known results about irreducibility of automorphic Galois representations and Langlands functoriality). In particular, when $F$ is a totally real field and $n$ is an odd prime number $\\leq 393$, we prove that (under certain automorphic conditions) the images of the residual representations $\\overline{\\rho}_{\\pi,\\ell}$ are as large as possible for infinitely many primes $\\ell$. In fact, we prove the large image conjecture (i.e. large image for almost all primes $\\ell$) when $F=\\mathbb{Q}$ and $n=5$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-06-04T22:00:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F80",
      "11F70",
      "20G40",
      "20G41"
    ],
    "keywords": [
      "global langlands correspondence",
      "cuspidal automorphic representations",
      "odd prime number",
      "automorphic galois representations",
      "lie type groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}