Fonctions L en géométrie rigide I: F-modules convergents ou surconvergents et conjecture de Dwork
Résumé
This article is the first one of a series of three articles devoted to L-functions. In this one we give a definition of the L-functions of convergent or overconvergent F- modules with the help of Teichmüller liftings and we establish the meromorphy of the L-functions of convergent F- modules in the closed unit disk. Whereas Wan established Dwork conjecture in a series of three articles , we give here a proof relying only on the generalyzed Monsky's trace formula of Wan and on an isogeny theorem of Katz. In the second article we'll give a definition of the L-functions of F-(iso)crystals by cohomological means and we'll show how it matches with the one given here: it gives back the one used in crystalline cohomology by Katz or Etesse, or the one used in rigid cohomology by Etesse-Le Stum, or the one used by Wan ; the aim is then to give a proof of Katz conjecture on p-adic unit roots and poles of these L-functions using rigid cohomology. In the third article we give an explicit form of these results for ordinary abelian schemes.
Origine : Fichiers produits par l'(les) auteur(s)