{
  "id": "1201.6016",
  "version": "v2",
  "published": "2012-01-29T06:35:02.000Z",
  "updated": "2012-07-25T14:58:27.000Z",
  "title": "Coleman-Gross height pairings and the $p$-adic sigma function",
  "authors": [
    "Jennifer S. Balakrishnan",
    "Amnon Besser"
  ],
  "comment": "AMS-LaTeX 17 pages",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "We give a direct proof that the Mazur-Tate and Coleman-Gross heights on elliptic curves coincide. The main ingredient is to extend the Coleman-Gross height to the case of divisors with non-disjoint support and, doing some $p$-adic analysis, show that, in particular, its component above $p$ gives, in the special case of an ordinary elliptic curve, the $p$-adic sigma function. We use this result to give a short proof of a theorem of Kim characterizing integral points on elliptic curves in some cases under weaker assumptions. As a further application, we give new formulas to compute double Coleman integrals from tangential basepoints.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2012-07-25T14:58:27.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05",
      "11S80"
    ],
    "keywords": [
      "adic sigma function",
      "coleman-gross height pairings",
      "elliptic curves coincide",
      "ordinary elliptic curve",
      "kim characterizing integral points"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1201.6016B"
    }
  }
}