{
  "id": "1711.02432",
  "version": "v1",
  "published": "2017-11-07T12:28:57.000Z",
  "updated": "2017-11-07T12:28:57.000Z",
  "title": "Computing the Cassels-Tate pairing on 3-isogeny Selmer groups via cubic norm equations",
  "authors": [
    "Monique van Beek",
    "Tom Fisher"
  ],
  "comment": "31 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We explain a method for computing the Cassels-Tate pairing on the 3-isogeny Selmer groups of an elliptic curve. This improves the upper bound on the rank of the elliptic curve coming from a descent by 3-isogeny, to that coming from a full 3-descent. One ingredient of our work is a new algorithm for solving cubic norm equations, that avoids the need for any S-unit computations. As an application, we show that the elliptic curves with torsion subgroup of order 3 and rank at least 13, found by Eroshkin, have rank exactly 13.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-11-07T12:28:57.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05",
      "11Y40"
    ],
    "keywords": [
      "selmer groups",
      "cassels-tate pairing",
      "solving cubic norm equations",
      "upper bound",
      "s-unit computations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}