{
  "id": "2002.05863",
  "version": "v1",
  "published": "2020-02-14T03:57:22.000Z",
  "updated": "2020-02-14T03:57:22.000Z",
  "title": "Rigid local systems and finite general linear groups",
  "authors": [
    "Nicholas M. Katz",
    "Pham Huu Tiep"
  ],
  "categories": [
    "math.RT",
    "math.NT"
  ],
  "abstract": "We use hypergeometric sheaves on $G_m/F_q$, which are particular sorts of rigid local systems, to construct explicit local systems whose arithmetic and geometric monodromy groups are the finite general linear groups $GL_n(q)$ for any $n \\ge 2$ and and any prime power $q$, so long as $q > 3$ when $n=2$. This paper continues a program of finding simple (in the sense of simple to remember) families of exponential sums whose monodromy groups are certain finite groups of Lie type, cf. [Gr], [KT1], [KT2], [KT3] for (certain) finite symplectic and unitary groups, or certain sporadic groups, cf. [KRL], [KRLT1], [KRLT2], [KRLT3]. The novelty of this paper is obtaining $GL_n(q)$ in this hypergeometric way. A pullback construction then yields local systems on $A^1/F_q$ whose geometric monodromy groups are $SL_n(q)$. These turn out to recover a construction of Abhyankar.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-02-14T03:57:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11T23",
      "20C33",
      "20G40"
    ],
    "keywords": [
      "finite general linear groups",
      "rigid local systems",
      "geometric monodromy groups",
      "construct explicit local systems",
      "yields local systems"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}