{
  "id": "2407.07193",
  "version": "v1",
  "published": "2024-07-09T19:23:16.000Z",
  "updated": "2024-07-09T19:23:16.000Z",
  "title": "Representation growth of Fuchsian groups and modular forms",
  "authors": [
    "Michael Larsen",
    "Jay Taylor",
    "Pham Huu Tiep"
  ],
  "comment": "25 pages",
  "categories": [
    "math.GR",
    "math.NT"
  ],
  "abstract": "Let $\\Gamma$ be a cocompact, oriented Fuchsian group which is not on an explicit finite list of possible exceptions and $q$ a sufficiently large prime power not divisible by the order of any non-trivial torsion element of $\\Gamma$. Then $|\\mathrm{Hom}(\\Gamma,\\mathrm{GL}_n(q))|\\sim c_{q,n} q^{(1-\\chi(\\Gamma))n^2}$, where $c_{q,n}$ is periodic in $n$. As a function of $q$, $c_{q,n}$ can be expressed as a Puiseux series in $1/q$ whose coefficients are periodic in $n$ and $q$. Moreover, this series is essentially the $q$-expansion of a meromorphic modular form of half-integral weight.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-07-09T19:23:16.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "20H10",
      "11F20",
      "11F27",
      "20C15",
      "20C33",
      "20G40"
    ],
    "keywords": [
      "representation growth",
      "meromorphic modular form",
      "sufficiently large prime power",
      "explicit finite list",
      "non-trivial torsion element"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}