{
  "id": "1702.01684",
  "version": "v1",
  "published": "2017-02-06T16:22:14.000Z",
  "updated": "2017-02-06T16:22:14.000Z",
  "title": "On the density of rational points on rational elliptic surfaces",
  "authors": [
    "Julie Desjardins"
  ],
  "comment": "31 pages, comments welcome",
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "abstract": "Let $\\mathscr{E}\\rightarrow\\mathbb{P}^1_\\mathbb{Q}$ be a non-trivial rational elliptic surface over $\\mathbb{Q}$ with base $\\mathbb{P}^1_\\mathbb{Q}$ (with a section). We show the Zariski-density of the rational points of $\\mathscr{E}$ when $\\mathscr{E}$ is isotrivial with non-zero $j$-invariant and when $\\mathscr{E}$ is non-isotrivial with a fiber of type $II^*$, $III^*$, $IV^*$ or $I^*_m$ ($m\\geq0$). Moreover, we produce examples of non-trivial elliptic surfaces whose rational points might not be dense. However, given the result at our disposal, we conjecture that any non-trivial elliptic surface has a dense set of $\\mathbb{Q}$-rational points.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-02-06T16:22:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14G05",
      "14J27",
      "14J26"
    ],
    "keywords": [
      "rational points",
      "non-trivial elliptic surface",
      "non-trivial rational elliptic surface",
      "produce examples",
      "dense set"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}