{
  "id": "math/0601580",
  "version": "v3",
  "published": "2006-01-24T12:10:06.000Z",
  "updated": "2008-03-17T11:39:27.000Z",
  "title": "Selmer groups of abelian varieties in extensions of function fields",
  "authors": [
    "Amilcar Pacheco"
  ],
  "comment": "final version, to appear in Mathematische Zeitschrift",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Let $k$ be a field of characteristic $q$, $\\cac$ a smooth geometrically connected curve defined over $k$ with function field $K:=k(\\cac)$. Let $A/K$ be a non constant abelian variety defined over $K$ of dimension $d$. We assume that $q=0$ or $>2d+1$. Let $p\\ne q$ be a prime number and $\\cac'\\to\\cac$ a finite geometrically \\textsc{Galois} and \\'etale cover defined over $k$ with function field $K':=k(\\cac')$. Let $(\\tau',B')$ be the $K'/k$-trace of $A/K$. We give an upper bound for the $\\bbz_p$-corank of the \\textsc{Selmer} group $\\text{Sel}_p(A\\times_KK')$, defined in terms of the $p$-descent map. As a consequence, we get an upper bound for the $\\bbz$-rank of the \\textsc{Lang-N\\'eron} group $A(K')/\\tau'B'(k)$. In the case of a geometric tower of curves whose \\textsc{Galois} group is isomorphic to $\\bbz_p$, we give sufficient conditions for the \\textsc{Lang-N\\'eron} group of $A$ to be uniformly bounded along the tower.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2008-03-17T11:39:27.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "function field",
      "selmer groups",
      "upper bound",
      "geometrically connected curve",
      "extensions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......1580P"
    }
  }
}