{
  "id": "0907.2529",
  "version": "v4",
  "published": "2009-07-15T08:44:56.000Z",
  "updated": "2010-01-07T19:05:28.000Z",
  "title": "Weak Approximation over Function Fields of Curves over Large or Finite Fields",
  "authors": [
    "Yong Hu"
  ],
  "comment": "numbering style changed; Theorem 2 in Section 1 strengthens its early version; many subsequent changes in Section 5",
  "journal": "Math. Ann. 348 (2010), No. 2, 357-377",
  "doi": "10.1007/s00208-010-0481-y",
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "abstract": "Let $K=k(C)$ be the function field of a curve over a field $k$ and let $X$ be a smooth, projective, separably rationally connected $K$-variety with $X(K)\\neq\\emptyset$. Under the assumption that $X$ admits a smooth projective model $\\pi: \\mathcal{X}\\to C$, we prove the following weak approximation results: (1) if $k$ is a large field, then $X(K)$ is Zariski dense; (2) if $k$ is an infinite algebraic extension of a finite field, then $X$ satisfies weak approximation at places of good reduction; (3) if $k$ is a nonarchimedean local field and $R$-equivalence is trivial on one of the fibers $\\mathcal{X}_p$ over points of good reduction, then there is a Zariski dense subset $W\\subseteq C(k)$ such that $X$ satisfies weak approximation at places in $W$. As applications of the methods, we also obtain the following results over a finite field $k$: (4) if $|k|>10$, then for a smooth cubic hypersurface $X/K$, the specialization map $X(K)\\longrightarrow \\prod_{p\\in P}\\mathcal{X}_p(\\kappa(p))$ at finitely many points of good reduction is surjective; (5) if $\\mathrm{char} k\\neq 2, 3$ and $|k|>47$, then a smooth cubic surface $X$ over $K$ satisfies weak approximation at any given place of good reduction.",
  "revisions": [
    {
      "version": "v4",
      "updated": "2010-01-07T19:05:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14M22",
      "14G05",
      "11G25",
      "14D10"
    ],
    "keywords": [
      "finite field",
      "function field",
      "satisfies weak approximation",
      "weak approximation results",
      "smooth cubic hypersurface"
    ],
    "tags": [
      "journal article"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2009arXiv0907.2529H"
    }
  }
}