{
  "id": "1811.09460",
  "version": "v1",
  "published": "2018-11-23T13:02:41.000Z",
  "updated": "2018-11-23T13:02:41.000Z",
  "title": "On Drinfeld modular forms of higher rank IV: Modular forms with level",
  "authors": [
    "Ernst-Ulrich Gekeler"
  ],
  "comment": "42 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We construct and study a natural compactification $\\overline{M}^r(N)$ of the moduli scheme $M^r(N)$ for rank-$r$ Drinfeld $\\F_q[T]$-modules with a structure of level $N \\in \\F_q[T]$. Namely, $\\overline{M}^r(N) = {\\rm Proj}\\,{\\bf Eis}(N)$, the projective variety associated with the graded ring ${\\bf Eis}(N)$ generated by the Eisenstein series of rank $r$ and level $N$. We use this to define the ring ${\\bf Mod}(N)$ of all modular forms of rank $r$ and level $N$. It equals the integral closure of ${\\bf Eis}(N)$ in their common quotient field $\\widetilde{\\MF}_r(N)$. Modular forms are characterized as those holomorphic functions on the Drinfeld space $\\Om^r$ with the right transformation behavior under the congruence subgroup $\\Ga(N)$ of $\\Ga = {\\rm GL}(r,\\F_q[T])$ (\"weak modular forms\") which, along with all their conjugates under $\\Ga/\\Ga(N)$, are bounded on the natural fundamental domain $\\BF$ for $\\Ga$ on $\\Om^r$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-11-23T13:02:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F52"
    ],
    "keywords": [
      "drinfeld modular forms",
      "higher rank",
      "common quotient field",
      "right transformation behavior",
      "weak modular forms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}