Rational points and fundamental groups: applications of the p -adic cohomology

Antoine Chambert-loir

Séminaire Bourbaki (2002-2003)

  • Volume: 45, page 125-146
  • ISSN: 0303-1179

Abstract

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I present results due to T. Ekedahl and H. Esnault concerning smooth projective varieties adefined over a field of positive characteristic, say  p , two points of which can be linked by a chain of rational curves. Examples are given by weakly unirational, or Fano varieties. Notably: 1) over a finite field, such varieties have a rational point, this generalizes the Chevalley-Warning Theorem; 2) over an algebraically closed field, the fundamental group of such varieties is finite and its order is prime to  p ; 3) over a finite field of cardinality  q , the number of rational points of two proper smooth varieties that are birational are congruent mod.  q . The proofs use the p -adic rigid cohomology defined by P. Berthelot.

How to cite

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Chambert-loir, Antoine. "Points rationnels et groupes fondamentaux : applications de la cohomologie $p$-adique." Séminaire Bourbaki 45 (2002-2003): 125-146. <http://eudml.org/doc/252128>.

@article{Chambert2002-2003,
abstract = {Je présenterai des résultats de T. Ekedahl et H. Esnault sur les variétés projectives lisses sur un corps de caractéristique strictement positive, disons $p$, dont deux points peuvent être liés par une chaîne de courbes rationnelles, par exemple faiblement unirationnelles, ou de Fano. Notamment : 1) sur un corps fini, de telles variétés ont un point rationnel, résultat qui généralise le théorème de Chevalley-Warning ; 2) sur un corps algébriquement clos, de telles variétés ont un groupe fondamental fini d’ordre premier à $p$ ; 3) sur un corps fini de cardinal $q$, deux variétés propres et lisses qui sont birationnelles ont même nombre de points rationnels modulo $q$. Les démonstrations utilisent la cohomologie rigide, $p$-adique, de P. Berthelot.},
author = {Chambert-loir, Antoine},
journal = {Séminaire Bourbaki},
keywords = {Fano varieties; chain rationaly connected varieties; rational points; fundamental group; rigid cohomology; slopes},
language = {fre},
pages = {125-146},
publisher = {Association des amis de Nicolas Bourbaki, Société mathématique de France},
title = {Points rationnels et groupes fondamentaux : applications de la cohomologie $p$-adique},
url = {http://eudml.org/doc/252128},
volume = {45},
year = {2002-2003},
}

TY - JOUR
AU - Chambert-loir, Antoine
TI - Points rationnels et groupes fondamentaux : applications de la cohomologie $p$-adique
JO - Séminaire Bourbaki
PY - 2002-2003
PB - Association des amis de Nicolas Bourbaki, Société mathématique de France
VL - 45
SP - 125
EP - 146
AB - Je présenterai des résultats de T. Ekedahl et H. Esnault sur les variétés projectives lisses sur un corps de caractéristique strictement positive, disons $p$, dont deux points peuvent être liés par une chaîne de courbes rationnelles, par exemple faiblement unirationnelles, ou de Fano. Notamment : 1) sur un corps fini, de telles variétés ont un point rationnel, résultat qui généralise le théorème de Chevalley-Warning ; 2) sur un corps algébriquement clos, de telles variétés ont un groupe fondamental fini d’ordre premier à $p$ ; 3) sur un corps fini de cardinal $q$, deux variétés propres et lisses qui sont birationnelles ont même nombre de points rationnels modulo $q$. Les démonstrations utilisent la cohomologie rigide, $p$-adique, de P. Berthelot.
LA - fre
KW - Fano varieties; chain rationaly connected varieties; rational points; fundamental group; rigid cohomology; slopes
UR - http://eudml.org/doc/252128
ER -

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