{
  "id": "2307.07593",
  "version": "v1",
  "published": "2023-07-14T19:33:37.000Z",
  "updated": "2023-07-14T19:33:37.000Z",
  "title": "Mod $\\ell$ gamma factors and a converse theorem for finite general linear groups",
  "authors": [
    "Jacksyn Bakeberg",
    "Mathilde Gerbelli-Gauthier",
    "Heidi Goodson",
    "Ashwin Iyengar",
    "Gilbert Moss",
    "Robin Zhang"
  ],
  "comment": "35 pages. Comments Welcome!",
  "categories": [
    "math.NT",
    "math.RT"
  ],
  "abstract": "For $q$ a power of a prime $p$, we study gamma factors of representations of $GL_n(\\mathbb{F}_q)$ over an algebraically closed field $k$ of positive characteristic $\\ell \\neq p$. We show that the reduction mod $\\ell$ of the gamma factor defined in characteristic zero fails to satisfy the analogue of the local converse theorem of Piatetski-Shapiro. To remedy this, we construct gamma factors valued in arbitrary $\\mathbb{Z}[1/p, \\zeta_p]$-algebras $A$, where $\\zeta_p$ is a primitive $p$-th root of unity, for Whittaker-type representations $\\rho$ and $\\pi$ of $GL_n(\\mathbb{F}_q)$ and $GL_m(\\mathbb{F}_q)$ over $A$. We let $P(\\pi)$ be the projective envelope of $\\pi$ and let $R(\\pi)$ be its endomorphism ring and define new gamma factors $\\widetilde\\gamma(\\rho \\times \\pi) = \\gamma((\\rho\\otimes_kR(\\pi)) \\times P(\\pi))$, which take values in the local Artinian $k$-algebra $R(\\pi)$. We prove a converse theorem for cuspidal representations using the new gamma factors. When $n=2$ and $m=1$ we construct a different ``new'' gamma factor $\\gamma^{\\ell}(\\rho,\\pi)$, which takes values in $k$ and satisfies a converse theorem.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-07-14T19:33:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20C33",
      "20C20",
      "11L05",
      "20G40",
      "11S40"
    ],
    "keywords": [
      "finite general linear groups",
      "characteristic zero fails",
      "construct gamma factors",
      "study gamma factors",
      "local converse theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}