{
  "id": "1010.4241",
  "version": "v3",
  "published": "2010-10-20T16:17:49.000Z",
  "updated": "2014-02-17T08:14:13.000Z",
  "title": "Semistable models for modular curves of arbitrary level",
  "authors": [
    "Jared Weinstein"
  ],
  "comment": "71 pages, 3 figures. Many errors corrected and details added following referee's suggestions",
  "categories": [
    "math.NT"
  ],
  "abstract": "We produce an integral model for the modular curve $X(Np^m)$ over the ring of integers of a sufficiently ramified extension of $\\mathbf{Z}_p$ whose special fiber is a {\\em semistable curve} in the sense that its only singularities are normal crossings. This is done by constructing a semistable covering (in the sense of Coleman) of the supersingular part of $X(Np^m)$, which is a union of copies of a Lubin-Tate curve. In doing so we tie together nonabelian Lubin-Tate theory to the representation-theoretic point of view afforded by Bushnell-Kutzko types. For our analysis it was essential to work with the Lubin-Tate curve not at level $p^m$ but rather at infinite level. We show that the infinite-level Lubin-Tate space (in arbitrary dimension, over an arbitrary nonarchimedean local field) has the structure of a perfectoid space, which is in many ways simpler than the Lubin-Tate spaces of finite level.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2014-02-17T08:14:13.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14G22",
      "22E50",
      "11F70"
    ],
    "keywords": [
      "modular curve",
      "arbitrary level",
      "semistable models",
      "arbitrary nonarchimedean local field",
      "lubin-tate curve"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 71,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1010.4241W"
    }
  }
}