Fonctions L en g\'eom\'etrie rigide I: F-modules convergents ou surconvergents et conjecture de Dwork

TL;DR

This paper defines L-functions for convergent and overconvergent F-modules, proves their meromorphy, and explores the limitations of slope filtrations and overconvergence in specific elliptic curve examples, contributing to the understanding of Dwork's conjecture.

Contribution

It introduces a new definition of L-functions for F-modules using Teichmüller liftings and demonstrates their meromorphicity, while providing explicit examples related to elliptic curves.

Findings

L-functions of convergent F-modules are meromorphic in the closed unit disk.

The slope filtration on an ordinary overconvergent F-module does not lift to an overconvergent filtration.

The unit-root sub-F-isocrystal of the de Rham cohomology of the Legendre family is not overconvergent.

Abstract

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 Teichm\"uller liftings and we establish the meromorphy of the L-functions of convergent F-modules in the closed unit disk. Wan has established Dwork conjecture in a series of three articles; owing to an isogeny theorem of Katz, his proof reduces to the ordinary case: here we prove, on two explicit examples related to families of elliptic curves, that the slope filtration on an ordinary overconvergent F-module does'nt lift to an overconvergent filtration. As a by-product we show that the unit-root sub-F-isocrystal of the de Rham cohomology of the Legendre family of ordinary elliptic curves is not overconvergent in Berthelot's sense. 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.