Images directes et fonctions L en cohomologie rigide

Jean-Yves Etesse (1)

1 IRMAR - Institut de Recherche Mathématique de Rennes

Let $k$ be a perfect field of characteristic $p>0$, $\mathcal{V}$ a complete discrete valuation ring with residue field $k$ and field of fractions $K$ of characteristic 0, and $S$ a separated $k$-scheme of finite type. When $S$ is smooth over $k$, we partially prove here a conjecture of Berthelot about the overconvergence of the higher direct images of the structure sheaf under a proper smooth morphism $f:X\rightarrow S$; when $k$ is perfect and $\mathcal{V}$ is tamely ramified such direct images are always convergent, not only for the structure sheaf but also for (almost) every convergent $F$-isocrystals. More generally, we prove this overconvergence when $f$ is liftable over $\mathcal{V}$, or when $X$ is a relative complete intersection in some projective spaces over $S$, and taking as coefficients any overconvergent isocrystals. We then apply these results to $L$-functions with coefficients such direct images with Frobenius structure: we derive rationality or meromorphy for these $L$-functions (Dwork's conjecture), and we study their $p$-adic unit zeroes and poles (Katz's conjecture) ; and explicit case concerns the ordinary abelian schemes. A more precise presentation of results by chapters is given in the introduction.

Mots clés: Algèbres de Monsky-Washnitzer . (F)-cristaux; (F)-isocristaux convergents ou surconvergents; cohomologie cristalline; cohomologie rigide; cohomologie étale; cohomologie syntomique; fonctions L; schéma abélien ordinaire; groupe p-divisible ordinaire; cristal de Dieudonné

Domaines: Géométrie algébrique [math.AG]

Jean-Yves Etesse. Images directes et fonctions L en cohomologie rigide. 2007. ⟨hal-00262316⟩