{
  "id": "1601.02373",
  "version": "v1",
  "published": "2016-01-11T09:49:21.000Z",
  "updated": "2016-01-11T09:49:21.000Z",
  "title": "On the divisibility by $p$ of the number of $F_{p}$-points of a variety",
  "authors": [
    "Lucile Devin"
  ],
  "comment": "20 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $X$ be an affine scheme of finite type over $\\mathbf{Z}$, we study the set $\\lbrace p\\in\\mathcal{P}, p\\nmid N_{X}(p)\\rbrace$ where $N_{X}(p)$ is the number of $\\mathbf{F}_{p}$-points of $X/\\mathbf{F}_{p}$. We prove a simple condition for the set to have positive lower-density in case $\\dim X\\leq 3$. Then we study the size of the smallest element of the set. We use sieve methods to bound the size of the least prime of $\\lbrace p\\in\\mathcal{P}, p\\nmid N_{X}(p)\\rbrace$ on average in particular families of hyperelliptic curves.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-01-11T09:49:21.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11R45",
      "11G25",
      "11N36"
    ],
    "keywords": [
      "divisibility",
      "finite type",
      "simple condition",
      "affine scheme",
      "smallest element"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 20,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160102373D"
    }
  }
}