Fonctions L associées aux F-isocristaux surconvergents II: Zéros et pôles unités

Abstract.

If X is a smooth variety over a finite field ? _q of characteristic p > 0 and ℱ is a p-adic sheaf associated to a representation of the fundamental group of X, N. Katz conjectures, in his Bourbaki talk 409, that the L function L (X, ℱ, t) has its p-adic unit roots and poles given in terms of p-adic étale cohomology. We prove this conjecture in the case of the structure sheaf ℱ = ℤ _p , that is for the Zeta function, and also more generally when the p-adic sheaf ℱ extends to a smooth sheaf on a compactification of X: as a consequence we get the Unit-Root Zeta function of Dwork and Sperber as an L function.

The idea of the proof is to get the p-adic étale cohomology with coefficients and compact support as the fixed points of Frobenius acting on rigid cohomology with compact support. For this purpose, we first build a crystalline Artin–Schreier short exact sequence on the syntomic site of a scheme which is separated of finite type over a perfect field k: this naturally generalizes the work of J.M. Fontaine and W. Messing in the proper smooth case. Then getting rigid cohomology with coefficients as a limit of crystalline cohomologies of variable level we deduce a long exact sequence connecting p-adic étale cohomology (with compact support) to rigid cohomology (with compact support). When X is smooth and affine over an algebraically closed field, the former exact sequence splits into short exact sequences that identify the p-adic étale cohomology with support of X to the part of its rigid cohomology invariant under Frobenius. We can then describe the p-adic unit roots and poles of the Zeta function of X; as a corallary we get the Unit-Root Zeta function of Dwork and Sperber as an L function.

In the appendix we show that the characteristic spaces of Frobenius in rigid cohomology commute with isometric extensions of the base, and that isocrystals associated to p-adic sheaves with finite monodromy are overconvergent: we thus obtain a p-adic proof of the rationality of the corresponding L-function.

Résumé

Nous démontrons la conjecture de Katz concernant la méromorphie et la caractérisation des zéros et pôles unités des fonctions L associées aux représentations p-adiques lorsque celles-ci se prolongent sur une compactification du schéma de base. Comme cas particuliers importants, on obtient celui de la fonction zêta d’un schéma quelconque et celui d’une représentation p-adique quelconque sur un schéma propre.