{
  "id": "math/0302138",
  "version": "v2",
  "published": "2003-02-12T10:02:01.000Z",
  "updated": "2004-10-26T07:55:55.000Z",
  "title": "Special points on products of modular curves",
  "authors": [
    "Bas Edixhoven"
  ],
  "comment": "21 pages, referee's remarks have been taken into account, some references updated, to appear in Duke Mathematical Journal",
  "journal": "Duke Math. J. 126 (2005), no. 2, 325--348.",
  "doi": "10.1215/S0012-7094-04-12624-7",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "We prove the Andre-Oort conjecture on special points of Shimura varieties for arbitrary products of modular curves, assuming the Generalized Riemann Hypothesis. More explicitly, this means the following. Let n be a positive integer, and let S be a subset of C^n (with C the complex numbers) consisting of points all of whose coordinates are j-invariants of elliptic curves with complex multiplications. Then we prove (under GRH) that the irreducible components of the Zariski closure of S are ``special subvarieties'', i.e., determined by isogeny conditions on coordinates and pairs of coordinates. A weaker variant is proved unconditionally.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2004-10-26T07:55:55.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14G35",
      "14K22",
      "11G15"
    ],
    "keywords": [
      "modular curves",
      "special points",
      "coordinates",
      "andre-oort conjecture",
      "isogeny conditions"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......2138E"
    }
  }
}