{
  "id": "1802.03204",
  "version": "v1",
  "published": "2018-02-09T11:10:47.000Z",
  "updated": "2018-02-09T11:10:47.000Z",
  "title": "The Betti map associated to a section of an abelian scheme (with an appendix by Z. Gao)",
  "authors": [
    "Yves André",
    "Pietro Corvaja",
    "Umberto Zannier"
  ],
  "comment": "31 pages, with an Appendix by Z. Gao",
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "abstract": "Given a point $\\xi$ on a complex abelian variety $A$, its abelian logarithm can be expressed as a linear combination of the periods of $A$ with real coefficients, the Betti coordinates of $\\xi$. When $(A, \\xi)$ varies in an algebraic family, these coordinates define a system of multivalued real-analytic functions. Computing its rank (in the sense of differential geometry) becomes important when one is interested about how often $\\xi$ takes a torsion value (for instance, Manin's theorem of the kernel implies that this coordinate system is constant in a family without fixed part only when $\\xi$ is a torsion section). We compute this rank in terms of the rank of a certain contracted form of the Kodaira-Spencer map associated to $(A, \\xi)$ (assuming $A$ without fixed part, and $\\mathbb{Z} \\xi$ Zariski-dense in $A$), and deduce some explicit lower bounds in special situations. For instance, we determine this rank in relative dimension $\\leq 3$, and study in detail the case of jacobians of families of hyperelliptic curves. Our main application, obtained in collaboration with Z. Gao, states that if $A\\to S$ is a principally polarized abelian scheme of relative dimension $g$ which has no non-trivial endomorphism (on any finite covering), and if the image of $S$ in the moduli space $\\mathcal{A}_g$ has dimension at least $g$, then the Betti map of any non-torsion section $\\xi$ is generically a submersion, so that $\\xi^{-1}A_{tors}$ is dense in $S(\\mathbb{C})$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-02-09T11:10:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "abelian scheme",
      "betti map",
      "relative dimension",
      "explicit lower bounds",
      "complex abelian variety"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 31,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}