{
  "id": "1711.02170",
  "version": "v1",
  "published": "2017-11-06T20:56:19.000Z",
  "updated": "2017-11-06T20:56:19.000Z",
  "title": "On Elliptic Curves of prime power conductor over imaginary quadratic fields with class number one",
  "authors": [
    "John Cremona",
    "Ariel Pacetti"
  ],
  "comment": "27 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "The main result of this paper is to generalize from $\\Q$ to each of the nine imaginary quadratic fields of class number one a result of Serre and Mestre-Oesterl\\'e of 1989, namely that if $E$ is an elliptic curve of prime conductor then either $E$ or a $2$-isogenous curve or a $3$-isogenous curve has prime discriminant. The proof is conditional in two ways: first that the curves are modular, so are associated to suitable Bianchi newforms; and secondly that a certain level-lowering conjecture holds for Bianchi newforms. We also classify all elliptic curves of prime power conductor and non-trivial torsion over each of the nine fields: in the case of $2$-torsion we find that such curves either have CM or with a small (finite) number of exceptions arise from a family analogous to the Setzer-Neumann family of elliptic curves over $\\Q$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-06T20:56:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05",
      "14H52"
    ],
    "keywords": [
      "prime power conductor",
      "imaginary quadratic fields",
      "elliptic curve",
      "class number",
      "bianchi newforms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}