{
  "id": "math/0305226",
  "version": "v1",
  "published": "2003-05-15T17:28:00.000Z",
  "updated": "2003-05-15T17:28:00.000Z",
  "title": "On the \"Section Conjecture\" in anabelian geometry",
  "authors": [
    "Jochen Koenigsmann"
  ],
  "comment": "21 pages, latex",
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "abstract": "Let X be a smooth projective curve of genus >1 over a field K which is finitely generated over the rationals. The section conjecture in Grothendieck's anabelian geometry says that the sections of the canonical projection from the arithmetic fundamental group of X onto the absolute Galois group of K are (up to conjugation) in one-to-one correspondence with K-rational points of X. The birational variant conjectures a similar correspondence where the fundamental group is replaced by the absolute Galois group of the function field K(X). The present paper proves the birational section conjecture for all X when K is replaced e.g. by the field of p-adic numbers. It disproves both conjectures for the field of real or p-adic algebraic numbers. And it gives a purely group theoretic characterization of the sections induced by K-rational points of X in the birational setting over almost arbitrary fields. As a biproduct we obtain Galois theoretic criteria for radical solvability of polynomial equations in more than one variable, and for a field to be PAC, to be large, or to be Hilbertian.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2003-05-15T17:28:00.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "12E30",
      "14H30"
    ],
    "keywords": [
      "absolute galois group",
      "grothendiecks anabelian geometry says",
      "k-rational points",
      "purely group theoretic characterization",
      "p-adic algebraic numbers"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2003math......5226K"
    }
  }
}