{
  "id": "1911.12551",
  "version": "v1",
  "published": "2019-11-28T06:24:38.000Z",
  "updated": "2019-11-28T06:24:38.000Z",
  "title": "Rational Points on Rational Curves",
  "authors": [
    "Brecken Beers",
    "Yih Sung"
  ],
  "comment": "10 pages",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "For a given elliptic curve, its associated $L$-function evaluated at $1$ is closely related to its real period. In this article, we generalize this principle to a rational curve. We count the rational points over all finite fields and use all the counting information to define two $L$-type series. Then we consider special values of these series at $1$. One of the $L$-type series matches the Dirichlet $L$-series of modulo $4$, so the evaluation at $1$ is $\\pi/4$; the special evaluation at $1$ of the other $L$-type series is equal to a real period associated to the rational curve. This identity confirms the general principle that an $L$-type series associated to a variety can reflect its geometry.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-11-28T06:24:38.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14G05",
      "11M41",
      "11F67"
    ],
    "keywords": [
      "rational curve",
      "rational points",
      "real period",
      "type series matches",
      "identity confirms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}