{
  "id": "1512.00773",
  "version": "v1",
  "published": "2015-12-02T16:54:04.000Z",
  "updated": "2015-12-02T16:54:04.000Z",
  "title": "On the arithmetic of abelian varieties",
  "authors": [
    "Mohamed Saidi",
    "Akio Tamagawa"
  ],
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "We prove some new results on the arithmetic of abelian varieties over function fields of one variable over finitely generated (infinite) fields. Among other things, we introduce certain new natural objects `discrete Selmer groups' and `discrete Shafarevich-Tate groups', and prove that they are finitely generated $\\Bbb Z$-modules. Further, we prove that in the isotrivial case, the discrete Shafarevich-Tate group vanishes and the discrete Selmer group coincides with the Mordell-Weil group. One of the key ingredients to prove these results is a new specialisation theorem \\`a la N\\'eron for first Galois cohomology groups, of the ($l$-adic) Tate module of abelian varieties which generalises N\\'eron's specialisation theorem for rational points of abelian varieties.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-12-02T16:54:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "abelian varieties",
      "arithmetic",
      "generalises nerons specialisation theorem",
      "discrete selmer group coincides",
      "first galois cohomology groups"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv151200773S"
    }
  }
}