{
  "id": "0811.3195",
  "version": "v1",
  "published": "2008-11-19T20:25:32.000Z",
  "updated": "2008-11-19T20:25:32.000Z",
  "title": "Analytic subvarieties with many rational points",
  "authors": [
    "Carlo Gasbarri"
  ],
  "comment": "44 pages. Submitted",
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "abstract": "We give a generalization of the classical Bombieri--Schneider--Lang criterion in transcendence theory. We give a local notion of $LG$--germ, which is similar to the notion of $E$-- function and Gevrey condition, and which generalize (and replace) the condition on derivatives in the theorem quoted above. Let $K\\subset \\Bbb C$ be a number field and $X$ a quasi--projective variety defined over $K$. Let $\\gamma\\colon M\\to X$ be an holomorphic map of finite order from a parabolic Riemann surface to $X$ such that the Zariski closure of the image of it is strictly bigger then one. Suppose that for every $p\\in X(K)\\cap\\gamma(M)$ the formal germ of $M$ near $P$ is an $LG$-- germ, then we prove that $X(K)\\cap\\gamma(M)$ is a finite set. Then we define the notion of conformally parabolic Kh\\\"aler varieties; this generalize the notion of parabolic Riemann surface. We show that on these varieties we can define a value distribution theory. The complementary of a divisor on a compact Kh\\\"aler manifold is conformally parabolic; in particular every quasi projective variety is. Suppose that $A$ is conformally parabolic variety of dimension $m$ over $\\Bbb C$ with Kh\\\"aler form $\\omega$ and $\\gamma\\colon A\\to X$ is an holomorphic map of finite order such that the Zariski closure of the image is strictly bigger then $m$. Suppose that for every $p\\in X(K)\\cap \\gamma (A)$, the image of $A$ is an $LG$--germ. then we prove that there exists a current $T$ on $A$ of bidegree $(1,1)$ such that $\\int_AT\\wedge\\omega^{m-1}$ explicitly bounded and with Lelong number bigger or equal then one on each point in $\\gamma^{-1}(X(K))$. In particular if $A$ is affine $\\gamma^{-1}(X(K))$ is not Zariski dense.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2008-11-19T20:25:32.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11J97",
      "14G40",
      "30D35"
    ],
    "keywords": [
      "rational points",
      "analytic subvarieties",
      "parabolic riemann surface",
      "conformally parabolic",
      "holomorphic map"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 44,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2008arXiv0811.3195G"
    }
  }
}