{
  "id": "2204.08551",
  "version": "v1",
  "published": "2022-04-18T20:33:47.000Z",
  "updated": "2022-04-18T20:33:47.000Z",
  "title": "Lower bound on the maximal number of rational points on curves over finite fields",
  "authors": [
    "Jonas Bergström",
    "Everett W. Howe",
    "Elisa Lorenzo García",
    "Christophe Ritzenthaler"
  ],
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "For a given genus $g \\geq 1$, we give minimal bounds for the maximal number of rational points on a smooth projective absolutely irreducible curve of genus $g$ over $\\F_q$. As a consequence of Katz-Sarnak theory, we first get for any given $g>0$, any $\\epsilon>0$ and all $q$ large enough, the existence of a curve of genus $g$ over $\\F_q$ with at least $1+q+ (2g-\\epsilon) \\sqrt{q}$ rational points. Then using sums of powers of traces of Frobenius of hyperelliptic curves, we get a lower bound of the form $1+q+1.71 \\sqrt{q}$ valid for $g \\geq 3$ and odd $q \\geq 11$. Finally, explicit constructions of towers of curves improve this result, with a bound of the form $1+q+4 \\sqrt{q} -32$ valid for all $g\\ge 2$ and for all~$q$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-18T20:33:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "rational points",
      "maximal number",
      "lower bound",
      "finite fields",
      "explicit constructions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}