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

Abstract

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.