{
  "id": "1801.02406",
  "version": "v1",
  "published": "2018-01-08T12:39:22.000Z",
  "updated": "2018-01-08T12:39:22.000Z",
  "title": "Tate's conjecture and the Tate-Shafarevich group over global function fields",
  "authors": [
    "Thomas H. Geisser"
  ],
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Let $\\mathcal X$ be a regular variety, flat and proper over a complete regular curve over a finite field, such that the generic fiber $X$ is smooth. We prove that the Brauer group of $\\mathcal X$ is finite if and only Tate's conjecture for divisors on $X$ holds and the Tate-Shafarevich group of the Albanese variety of $X$ is finite, generalizing a theorem of Artin and Grothendieck for surfaces to arbitrary relative dimension.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-01-08T12:39:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "global function fields",
      "tates conjecture",
      "tate-shafarevich group",
      "complete regular curve",
      "finite field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}