{
  "id": "1910.06257",
  "version": "v1",
  "published": "2019-10-14T16:34:59.000Z",
  "updated": "2019-10-14T16:34:59.000Z",
  "title": "Quantitative arithmetic of diagonal degree $2$ K3 surfaces",
  "authors": [
    "Damián Gvirtz",
    "Daniel Loughran",
    "Masahiro Nakahara"
  ],
  "comment": "52 pages",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "In this paper we study the existence of rational points for the family of K3 surfaces over $\\mathbb{Q}$ given by $$w^2 = A_1x_1^6 + A_2x_2^6 + A_3x_3^6.$$ When the coefficients are ordered by height, we show that the Brauer group is almost always trivial, and find the exact order of magnitude of surfaces for which there is a Brauer-Manin obstruction to the Hasse principle. Our results show definitively that K3 surfaces can have a Brauer-Manin obstruction to the Hasse principle that is only explained by odd order torsion.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-10-14T16:34:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14G05",
      "14F22"
    ],
    "keywords": [
      "k3 surfaces",
      "diagonal degree",
      "quantitative arithmetic",
      "hasse principle",
      "brauer-manin obstruction"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 52,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}