{
  "id": "math/9809208",
  "version": "v1",
  "published": "1998-09-03T00:00:00.000Z",
  "updated": "1998-09-03T00:00:00.000Z",
  "title": "Steinitz class of Mordell groups of elliptic curves with complex multiplication",
  "authors": [
    "Tong Liu",
    "Xianke Zhang"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Let E be an elliptic curve having Complex Multiplication by the full ring O_K of integers of K=Q(\\sqrt{-D}), let H=K(j(E)) be the Hilbert class field of K. Then the Mordell-Weil group E(H) is an O_K-module, and its structure denpends on its Steinitz class St(E), which is studied here. In partucular, when D is a prime number, it is proved that St(E)=1 if D\\equiv 3 (mod 4); and St(E)=[P]^t if D\\equiv 1 (mod 4), where [P] is the ideal class of K represented by prime factor P of 2 in K, t is a fixed integer. General structures are also discussed for St(E) and for modules over Dedekind domain. These results develop the results by D. Dummit and W. Miller for D=10 and some elliptic curves to more general D and general elliptic curves.",
  "revisions": [
    {
      "version": "v1",
      "updated": "1998-09-03T00:00:00.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "complex multiplication",
      "mordell groups",
      "general elliptic curves",
      "steinitz class st",
      "hilbert class field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1998math......9208L"
    }
  }
}