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

الأصل · EN

In this talk, I report on three theorems concerning algebraic varieties over a field of characteristic p>0. a) over a finite field of cardinal q, two proper smooth varieties which are geometrically birational have the same number of rational points modulo q (cf. Ekedahl, 1983). b) over a finite field of cardinal q, a proper smooth variety which is rationally chain connected, or Fano, or weakly unirational, has a number of rational points congruent to 1 modulo q (Esnault, 2003). c) over an algebraic closed field of caracteristic p>0, the fundamental group of a proper smooth variety which is rationally chain connected, or Fano, or weakly unirational, is a finite group of order prime to p (cf. Ekedahl, 1983). The common feature of the proofs is a control of the p-adic valuations of Frobenius and is best explained within the framework of Berthelot's rigid cohomology. I also explain its relevant properties.