{
  "id": "1312.7394",
  "version": "v2",
  "published": "2013-12-28T05:22:51.000Z",
  "updated": "2015-02-03T21:17:02.000Z",
  "title": "Topological modular forms with level structure",
  "authors": [
    "Michael Hill",
    "Tyler Lawson"
  ],
  "comment": "53 pages. Heavily revised, including the addition of a new section on background tools from homotopy theory",
  "categories": [
    "math.AT",
    "math.NT"
  ],
  "abstract": "The cohomology theory known as Tmf, for \"topological modular forms,\" is a universal object mapping out to elliptic cohomology theories, and its coefficient ring is closely connected to the classical ring of modular forms. We extend this to a functorial family of objects corresponding to elliptic curves with level structure and modular forms on them. Along the way, we produce a natural way to restrict to the cusps, providing multiplicative maps from Tmf with level structure to forms of K-theory. In particular, this allows us to construct a connective spectrum tmf_0(3) consistent with properties suggested by Mahowald and Rezk. This is accomplished using the machinery of logarithmic structures. We construct a sheaf of locally even-periodic elliptic cohomology theories, equipped with highly structured multiplication, on the log-\\'etale site of the moduli of elliptic curves. Evaluating this sheaf on modular curves produces Tmf with level structure.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2013-12-28T05:22:51.000Z",
      "comment": "42 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-02-03T21:17:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "55N34",
      "55P43",
      "11F23",
      "11G18",
      "14F20"
    ],
    "keywords": [
      "topological modular forms",
      "level structure",
      "cohomology theory",
      "elliptic curves",
      "modular curves produces tmf"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 53,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2013arXiv1312.7394H"
    }
  }
}