{
  "id": "1201.0252",
  "version": "v1",
  "published": "2011-12-31T12:22:42.000Z",
  "updated": "2011-12-31T12:22:42.000Z",
  "title": "Ranks of elliptic curves with prescribed torsion over number fields",
  "authors": [
    "Johan Bosman",
    "Peter Bruin",
    "Andrej Dujella",
    "Filip Najman"
  ],
  "comment": "25 pages",
  "journal": "Int. Math. Res. Not. IMRN 2014 (11) (2014), 2885-2923",
  "doi": "10.1093/imrn/rnt013",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "We study the structure of the Mordell--Weil group of elliptic curves over number fields of degree 2, 3, and 4. We show that if $T$ is a group, then either the class of all elliptic curves over quadratic fields with torsion subgroup $T$ is empty, or it contains curves of rank~0 as well as curves of positive rank. We prove a similar but slightly weaker result for cubic and quartic fields. On the other hand, we find a group $T$ and a quartic field $K$ such that among the elliptic curves over $K$ with torsion subgroup $T$, there are curves of positive rank, but none of rank~0. We find examples of elliptic curves with positive rank and given torsion in many previously unknown cases. We also prove that all elliptic curves over quadratic fields with a point of order 13 or 18 and all elliptic curves over quartic fields with a point of order 22 are isogenous to one of their Galois conjugates and, by a phenomenon that we call \\emph{false complex multiplication}, have even rank. Finally, we discuss connections with elliptic curves over finite fields and applications to integer factorization.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-12-31T12:22:42.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "elliptic curves",
      "number fields",
      "prescribed torsion",
      "quartic field",
      "positive rank"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1201.0252B"
    }
  }
}