{
  "id": "1108.2526",
  "version": "v1",
  "published": "2011-08-11T21:52:10.000Z",
  "updated": "2011-08-11T21:52:10.000Z",
  "title": "The distribution of the number of points on trigonal curves over $\\F_q$",
  "authors": [
    "Melanie Matchett Wood"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "We give a short determination of the distribution of the number of $\\F_q$-rational points on a random trigonal curve over $\\F_q$, in the limit as the genus of the curve goes to infinity. In particular, the expected number of points is $q+2-\\frac{1}{q^2+q+1}$, contrasting with recent analogous results for cyclic $p$-fold covers of $\\mathbb P^1$ and plane curves which have an expected number of points of $q+1$ (by work of Kurlberg, Rudnick, Bucur, David, Feigon and Lal\\'in) and curves which are complete intersections which have an expected number of points $<q+1$ (by work of Bucur and Kedlaya). We also give a conjecture for the expected number of points on a random $n$-gonal curve with full $S_n$ monodromy based on function field analogs of Bhargava's heuristics for counting number fields.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-08-11T21:52:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "expected number",
      "distribution",
      "random trigonal curve",
      "function field analogs",
      "counting number fields"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1108.2526M"
    }
  }
}