{
  "id": "1210.3299",
  "version": "v2",
  "published": "2012-10-11T17:05:10.000Z",
  "updated": "2013-01-29T15:39:11.000Z",
  "title": "The Tate-Voloch Conjecture in a Power of a Modular Curve",
  "authors": [
    "Philipp Habegger"
  ],
  "comment": "Corrected some typos in version 2",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $p$ be a prime. Tate and Voloch proved that a point of finite order in the algebraic torus cannot be $p$-adically too close to a fixed subvariety without lying on it. The current work is motivated by the analogy between torsion points on semi-abelian varieties and special or CM points on Shimura varieties. We prove the analog of Tate and Voloch's result in a power of the modular curve Y(1) on replacing torsion points by points corresponding to a product of elliptic curves with complex multiplication and ordinary reduction. Moreover, we show that the assumption on ordinary reduction is necessary.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2013-01-29T15:39:11.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G18",
      "14G35",
      "11J61"
    ],
    "keywords": [
      "modular curve",
      "tate-voloch conjecture",
      "ordinary reduction",
      "cm points",
      "algebraic torus"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1210.3299H"
    }
  }
}