{
  "id": "2006.14907",
  "version": "v1",
  "published": "2020-06-26T10:48:31.000Z",
  "updated": "2020-06-26T10:48:31.000Z",
  "title": "Explicit uniform bounds for Brauer groups of singular K3 surfaces",
  "authors": [
    "Francesca Balestrieri",
    "Alexis Johnson",
    "Rachel Newton"
  ],
  "comment": "37 pages, comments welcome!",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Let $k$ be a number field. We give an explicit bound, depending only on $[k:\\mathbf{Q}]$, on the size of the Brauer group of a K3 surface $X/k$ that is geometrically isomorphic to the Kummer surface attached to a product of CM elliptic curves. As an application, we show that the Brauer-Manin set for such a variety is effectively computable. In addition, we prove an effective version of the strong Shafarevich conjecture for singular K3 surfaces by giving an explicit bound, depending only on $[k:\\mathbf{Q}]$, on the number of $\\mathbf{C}$-isomorphism classes of singular K3 surfaces defined over $k$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-26T10:48:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14F22",
      "14J28",
      "14G05"
    ],
    "keywords": [
      "singular k3 surfaces",
      "explicit uniform bounds",
      "brauer group",
      "explicit bound",
      "cm elliptic curves"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 37,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}