{
  "id": "1005.4206",
  "version": "v2",
  "published": "2010-05-23T14:17:52.000Z",
  "updated": "2010-05-27T12:15:28.000Z",
  "title": "The Tate-Shafarevich group for elliptic curves with complex multiplication II",
  "authors": [
    "J. Coates",
    "Z. Liang",
    "R. Sujatha"
  ],
  "comment": "24 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let E be an elliptic curve over Q with complex multiplication. The aim of the present paper is to strengthen the theoretical and numerical results of \\cite{CZS}. For each prime p, let t_{E/Q, p} denote the Z_p-corank of the p-primary subgroup of the Tate-Shafarevich group of E/Q. For each \\epsilon 0, we prove that t_{E/Q, p} is bounded above by (1/2+\\epsilon)p for all sufficiently large good ordinary primes p. We also do numerical calculations on one such E of rank 3, and 5 such E of rank 2, showing in all cases that t_{E/Q, p} = 0 for all good ordinary primes p < 30,000. In fact, we show that, with the possible exception of one good ordinary prime in this range for just one of the curves of rank 2, the p-primary subgroup of the Tate-Shafarevich group of the curve is zero (always supposing p is a good ordinary prime).",
  "revisions": [
    {
      "version": "v2",
      "updated": "2010-05-27T12:15:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05",
      "11G15"
    ],
    "keywords": [
      "tate-shafarevich group",
      "elliptic curve",
      "complex multiplication",
      "ordinary prime",
      "p-primary subgroup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 24,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1005.4206C"
    }
  }
}