{
  "id": "2011.01186",
  "version": "v1",
  "published": "2020-11-02T18:22:02.000Z",
  "updated": "2020-11-02T18:22:02.000Z",
  "title": "A positive proportion of cubic fields are not monogenic yet have no local obstruction to being so",
  "authors": [
    "Levent Alpöge",
    "Manjul Bhargava",
    "Ari Shnidman"
  ],
  "comment": "13 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We show that a positive proportion of cubic fields are not monogenic, despite having no local obstruction to being monogenic. Our proof involves the comparison of $2$-descent and $3$-descent in a certain family of Mordell curves $E_k \\colon y^2 = x^3 + k$. As a by-product of our methods, we show that, for every $r \\geq 0$, a positive proportion of curves $E_k$ have Tate--Shafarevich group with $3$-rank at least $r$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-11-02T18:22:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R16",
      "11R45",
      "11G05"
    ],
    "keywords": [
      "positive proportion",
      "local obstruction",
      "cubic fields",
      "mordell curves",
      "tate-shafarevich group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}