Points rationnels et groupes fondamentaux : applications de la cohomologie $p$-adique

Points rationnels et groupes fondamentaux : applications de la cohomologie

p

-adique

Chambert-loir, Antoine

Séminaire Bourbaki : volume 2002/2003, exposés 909-923, Astérisque, no. 294 (2004), Exposé no. 914, pp. 125-146.

Résumé
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.

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.

MR Zbl | 1 citation dans Numdam

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. https://www.numdam.org/item/SB_2002-2003__45__125_0/

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