{
  "id": "1802.07290",
  "version": "v1",
  "published": "2018-02-20T19:15:15.000Z",
  "updated": "2018-02-20T19:15:15.000Z",
  "title": "Growth of the analytic rank of rational elliptic curves over quintic fields",
  "authors": [
    "Michele Fornea"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "Given a rational elliptic curve $E_{/\\mathbb{Q}}$, we denote by $G_5(E;X)$ the number of quintic fields of absolute discriminant at most $X$ such that the analytic rank of $E$ grows over $K$, i.e., $\\mathrm{r}_\\mathrm{an}(E/K)>\\mathrm{r}_\\mathrm{an}(E/\\mathbb{Q})$. We show that $G_5(E;X)\\asymp_{+\\infty} X$ when the elliptic curve $E$ has odd conductor and at least one prime of multiplicative reduction. As Bhargava showed that the number of quintic fields of absolute discriminant at most $X$ is asymptotic to $c_5 X$, for some positve constant $c_5>0$, our result exposes the growth of the analytic rank as a very common circumstance over quintic fields.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-02-20T19:15:15.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "rational elliptic curve",
      "quintic fields",
      "analytic rank",
      "absolute discriminant",
      "odd conductor"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}