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.

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.

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