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This article is the first one of a series of three articles devoted toL-functions. In this one we give a definition of the L-functions of convergentor overconvergent F-modules with the help of Teichm\"uller liftings and weestablish the meromorphy of the L-functions of convergent F-modules in theclosed unit disk. Wan has established Dwork conjecture in a series of threearticles; owing to an isogeny theorem of Katz, his proof reduces to theordinary case: here we prove, on two explicit examples related to families ofelliptic curves, that the slope filtration on an ordinary overconvergentF-module does'nt lift to an overconvergent filtration. As a by-product we showthat the unit-root sub-F-isocrystal of the de Rham cohomology of the Legendrefamily of ordinary elliptic curves is not overconvergent in Berthelot's sense.In the second article we'll give a definition of the L-functions ofF-(iso)crystals by cohomological means and we'll show how it matches with theone given here: it gives back the one used in crystalline cohomology by Katz orEtesse, or the one used in rigid cohomology by Etesse-Le Stum, or the one usedby Wan ; the aim is then to give a proof of Katz conjecture on p-adic unitroots and poles of these L-functions using rigid cohomology. In the thirdarticle we give an explicit form of these results for ordinary abelian schemes.

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