{
  "id": "1209.3527",
  "version": "v3",
  "published": "2012-09-16T22:54:59.000Z",
  "updated": "2015-01-26T02:26:34.000Z",
  "title": "K3 surfaces and equations for Hilbert modular surfaces",
  "authors": [
    "Noam Elkies",
    "Abhinav Kumar"
  ],
  "comment": "83 pages. Final version",
  "journal": "Algebra and Number Theory 8:10 (2014), 2297-2411",
  "doi": "10.2140/ant.2014.8.2297",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "We outline a method to compute rational models for the Hilbert modular surfaces Y_{-}(D), which are coarse moduli spaces for principally polarized abelian surfaces with real multiplication by the ring of integers in Q(sqrt{D}), via moduli spaces of elliptic K3 surfaces with a Shioda-Inose structure. In particular, we compute equations for all thirty fundamental discriminants D with 1 < D < 100, and analyze rational points and curves on these Hilbert modular surfaces, producing examples of genus-2 curves over Q whose Jacobians have real multiplication over Q.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-11-11T06:42:58.000Z",
      "comment": "83 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v3",
      "updated": "2015-01-26T02:26:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F41",
      "14G35",
      "14J28",
      "14J27"
    ],
    "keywords": [
      "hilbert modular surfaces",
      "real multiplication",
      "coarse moduli spaces",
      "thirty fundamental discriminants",
      "analyze rational points"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 83,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1209.3527E"
    }
  }
}