{
  "id": "0806.4267",
  "version": "v3",
  "published": "2008-06-26T09:45:05.000Z",
  "updated": "2009-09-02T19:51:47.000Z",
  "title": "On the vanishing of Selmer groups for elliptic curves over ring class fields",
  "authors": [
    "M. Longo",
    "S. Vigni"
  ],
  "comment": "31 pages, minor modifications; final version, to appear in Journal of Number Theory",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Let E be a rational elliptic curve of conductor N without complex multiplication and let K be an imaginary quadratic field of discriminant D prime to N. Assume that the number of primes dividing N and inert in K is odd, and let H be the ring class field of K of conductor c prime to ND with Galois group G over K. Fix a complex character \\chi of G. Our main result is that if the special value of the \\chi-twisted L-function of E/K is non-zero then the tensor product (with respect to \\chi) of the p-Selmer group of E/H with W over Z[G] is 0 for all but finitely many primes p, where W is a suitable finite extension of Z_p containing the values of \\chi. Our work extends results of Bertolini and Darmon to almost all non-ordinary primes p and also offers alternative proofs of a \\chi-twisted version of the Birch and Swinnerton-Dyer conjecture for E over H (Bertolini and Darmon) and of the vanishing of the p-Selmer group of E/K for almost all p (Kolyvagin) in the case of analytic rank zero.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2009-09-02T19:51:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05",
      "11G40"
    ],
    "keywords": [
      "ring class field",
      "selmer groups",
      "p-selmer group",
      "rational elliptic curve",
      "imaginary quadratic field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0806.4267L"
    }
  }
}