{
  "id": "2102.00900",
  "version": "v1",
  "published": "2021-02-01T15:17:25.000Z",
  "updated": "2021-02-01T15:17:25.000Z",
  "title": "Curves of fixed gonality with many rational points",
  "authors": [
    "Floris Vermeulen"
  ],
  "comment": "9 pages",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Given a gonality $\\gamma$ and a prime power $q\\geq \\gamma -1$ we show that for every large genus $g$ there exists a curve $C$ defined over $\\mathbb{F}_q$ of genus $g$ and gonality $\\gamma$ and with exactly $\\gamma(q+1)$ $\\mathbb{F}_q$-rational points. This is the maximal number of rational points allowed. This answers a recent conjecture by Faber--Grantham. Our methods are based on Poonen's work on squarefree values of polynomials together with a Newton polygon argument.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-02-01T15:17:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "rational points",
      "fixed gonality",
      "newton polygon argument",
      "large genus",
      "maximal number"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 9,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}