{
  "id": "math/0702593",
  "version": "v3",
  "published": "2007-02-20T21:58:13.000Z",
  "updated": "2008-10-02T15:27:14.000Z",
  "title": "Analytic curves in algebraic varieties over number fields",
  "authors": [
    "Jean-Benoît Bost",
    "Antoine Chambert-Loir"
  ],
  "comment": "55 pages. To appear in \"Algebra, Arithmetic, and Geometry: In Honor of Y.i. Manin\", Y. Tschinkel & Yu. Manin editors, Birkh\\\"auser, 2009",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "We establish algebraicity criteria for formal germs of curves in algebraic varieties over number fields and apply them to derive a rationality criterion for formal germs of functions, which extends the classical rationality theorems of Borel-Dwork and P\\'olya-Bertrandias valid over the projective line to arbitrary algebraic curves over a number field. The formulation and the proof of these criteria involve some basic notions in Arakelov geometry, combined with complex and rigid analytic geometry (notably, potential theory over complex and $p$-adic curves). We also discuss geometric analogues, pertaining to the algebraic geometry of projective surfaces, of these arithmetic criteria.",
  "revisions": [
    {
      "version": "v3",
      "updated": "2008-10-02T15:27:14.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14G40",
      "31A15",
      "14G22"
    ],
    "keywords": [
      "number field",
      "algebraic varieties",
      "analytic curves",
      "formal germs",
      "rigid analytic geometry"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 55,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007math......2593B"
    }
  }
}