{
  "id": "1602.03140",
  "version": "v1",
  "published": "2016-02-09T19:53:25.000Z",
  "updated": "2016-02-09T19:53:25.000Z",
  "title": "Serre's problem on the density of isotropic fibres in conic bundles",
  "authors": [
    "Efthymios Sofos"
  ],
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Let $\\pi:X\\to \\mathbb{P}^1_{\\mathbb{Q}}$ be a non-singular conic bundle over $\\mathbb{Q}$ having $n$ non-split fibres and denote by $N(\\pi,B)$ the cardinality of the fibres of Weil height at most $B$ that possess a rational point. Serre showed in $1990$ that a direct application of the large sieve yields $$N(\\pi,B)\\ll B^2(\\log B)^{-n/2}$$ and raised the problem of proving that this is the true order of magnitude of $N(\\pi,B)$. We solve this problem for all non-singular conic bundles of rank at most $3$. Our method comprises the use of Hooley neutralisers, estimating divisor sums over values of binary forms, and an application of the Rosser-Iwaniec sieve.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-02-09T19:53:25.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14G05",
      "14D10",
      "11N36",
      "11G35"
    ],
    "keywords": [
      "serres problem",
      "isotropic fibres",
      "non-singular conic bundle",
      "large sieve yields",
      "non-split fibres"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2016arXiv160203140S"
    }
  }
}