For a $\nabla$-module M over the ring $K[[x]}_0$ of bounded functions over a p-adic local field K we define the notion of special and generic log-growth filtrations oil the base of the power series development of the solutions and horizontal sections. Moreover, if M also admits a Frobenius structure then it is endowed with generic and special Frobenius slope filtrations. We will show that in the case of M a $\phi-\nabla$-module of rank 2, the Frobenius polygon for M and the log-growth polygon for its dual, $M^\star$, coincide, this is proved by showing explicit relationships between the filtrations. This will lead us to formulate some conjectural links between the behaviours of the filtrations arising from the log-growth and Frobenius structures of the differential module. This coincidence between the two polygons was only known for the hypergeometric cases by Dwork

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The naive analogue of the Néron–Ogg–Shafarevich criterion is false for K3 surfaces, that is, there exist K3 surfaces over Henselian, discretely valued fields K, with unramified ℓ‐adic étale cohomology groups, but which do not admit good reduction over K. Assuming potential semi‐stable reduction, we show how to correct this by proving that a K3 surface has good reduction if and only if Hnormalét2false(XK¯,Qℓfalse) is unramified, and the associated Galois representation over the residue field coincides with the second cohomology of a certain ‘canonical reduction’ of X. We also prove the corresponding results for p‐adic étale cohomology.

Weights in rigid cohomology applications to unipotent F-isocrystals Annales scientifiques de l'É.N.S. 4 e série, tome 31, n o 5 (1998), p. 683-715 © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1998, tous droits réservés. L'accès aux archives de la revue « Annales scientifiques de l'É.N.S. » (http://www. elsevier.com/locate/ansens) implique l'accord avec les conditions générales d'utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d'une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/

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