Points rationnels et groupes fondamentaux : applications de la cohomologie p-adique
Séminaire Bourbaki : volume 2002/2003, exposés 909-923, Astérisque, no. 294 (2004), Exposé no. 914, pp. 125-146.

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.

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.

Classification : 14M20, 14J45, 14G15, 14G05, 14H30, 14Cxx, 14F30
Mot clés : variété de Fano, variété rationnellement connexes par chaînes, points rationnels, groupe fondamental, cohomologie rigide, pentes
Keywords: Fano varieties, chain rationaly connected varieties, rational points, fundamental group, rigid cohomology, slopes
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Chambert-loir, Antoine. Points rationnels et groupes fondamentaux : applications de la cohomologie $p$-adique, dans Séminaire Bourbaki : volume 2002/2003, exposés 909-923, Astérisque, no. 294 (2004), Exposé no. 914, pp. 125-146. http://archive.numdam.org/item/SB_2002-2003__45__125_0/

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