{
  "id": "1909.04098",
  "version": "v1",
  "published": "2019-09-09T18:50:09.000Z",
  "updated": "2019-09-09T18:50:09.000Z",
  "title": "Growth of points on hyperelliptic curves",
  "authors": [
    "Christopher Keyes"
  ],
  "comment": "12 pages. Comments welcome",
  "categories": [
    "math.NT"
  ],
  "abstract": "Fix a hyperelliptic curve $C/\\mathbb{Q}$ of genus $g$, and consider the number fields $K/\\mathbb{Q}$ generated by the algebraic points of $C$. In this paper, we study the number of such extensions with fixed degree $n$ and discriminant bounded by $X$. We show that when $g \\geq 1$ and $n$ is at least the degree of $C$, with $n$ even if the degree of $C$ is even, there are $\\gg X^{c_n}$ such extensions, where $c_n$ is a positive constant depending on $g$ which tends to $1/4$ as $n \\to \\infty$. This builds on work of Lemke Oliver and Thorne, who in the case where $C$ is an elliptic curve put lower bounds on the number of extensions with fixed degree and bounded discriminant over which the rank of $C$ grows with specified root number.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-09-09T18:50:09.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "hyperelliptic curve",
      "extensions",
      "fixed degree",
      "lemke oliver",
      "algebraic points"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}