{
  "id": "math/0210173",
  "version": "v2",
  "published": "2002-10-11T15:00:45.000Z",
  "updated": "2004-07-23T13:04:19.000Z",
  "title": "Computing the cardinality of CM elliptic curves using torsion points",
  "authors": [
    "F. Morain"
  ],
  "comment": "Revised and shortened version, including more material using discriminants of curves and division polynomials",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let E be an elliptic curve having complex multiplication by a given quadratic order of an imaginary quadratic field K. The field of definition of E is the ring class field Omega of the order. If the prime p splits completely in Omega, then we can reduce E modulo one the factors of p and get a curve Ep defined over GF(p). The trace of the Frobenius of Ep is known up to sign and we need a fast way to find this sign. For this, we propose to use the action of the Frobenius on torsion points of small order built with class invariants a la Weber, in a manner reminiscent of the Schoof-Elkies-Atkin algorithm for computing the cardinality of a given elliptic curve modulo p. We apply our results to the Elliptic Curve Primality Proving algorithm (ECPP).",
  "revisions": [
    {
      "version": "v2",
      "updated": "2004-07-23T13:04:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G15",
      "11G20"
    ],
    "keywords": [
      "cm elliptic curves",
      "torsion points",
      "cardinality",
      "elliptic curve primality proving algorithm",
      "ring class field omega"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math.....10173M"
    }
  }
}