We define two decreasing filtrations by ramification groups on the absolute Galois group of a complete discrete valuation field whose residue field may not be perfect. In the classical case where the residue field is perfect, we recover the classical upper numbering filtration. The definition uses rigid geometry and log-structures. We also establish some of their properties.

The p-adic Simpson correspondence, recently initiated by Gerd Faltings, aims at describing all p-adic representations of the fundamental group of a proper smooth variety over a p-adic field in terms of linear algebra—namely Higgs bundles. This book undertakes a systematic development of the theory following two new approaches. It mainly focuses on generalized representations of the fundamental group that are p-adically close to the trivial representation. The first approach relies on a new family of period rings built from the torsor of deformations of the variety over a universal p-adic thickening defined by J. M. Fontaine. The second approach introduces a crystalline-type topos and replaces the notion of Higgs bundles with that of Higgs isocrystals. The book shows the compatibility of the two constructions and the compatibility of the correspondence with the natural cohomologies. The last part of the book contains results of wider interest in p-adic Hodge theory. The reader will find a concise introduction to Faltings' theory of almost étale extensions and a chapter devoted to the Faltings topos. Though this topos is the general framework for Faltings' approach in p-adic Hodge theory, it remains relatively unexplored.

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RésuméNous développons, pour un faisceauétale -adique sur un trait complet de caractéristique p > 0, la notion de variété caractéristique. Notre approche, inspirée de l'analyse micro-locale de Kashiwara et Schapira, est un pendant faisceautique de notre théorie de ramification des corps locauxà corps résiduel quelconque. Nous présentons la principale propriété que devrait satisfaire la variété caractéristique (conjecture de l'isogénie), et nous la démontrons pour les faisceaux de rang un sans restriction sur le trait, ou inconditionnellement sur le faisceau si le corps résiduel du trait est parfait. AbstractWe develop, for an -adicétale sheaf on a complete trait of characteristic p > 0, the notion of characteristic variety. Our approach, inspired by the microlocal analysis of Kashiwara and Schapira, is a complement to our ramification theory for local fields with general residue fields. We formulate the main property that should be satisfied by the characteristic variety (the isogeny conjecture), and prove it for rank one sheaves unconditionally on the trait, or unconditionally on the sheaf if the residue field of the trait is perfect.

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Laumon introduced the local Fourier transform for -adic Galois representations of local fields, of equal characteristic p different from , as a powerful tool for studying the Fourier-Deligne transform of -adic sheaves over the affine line. In this article, we compute explicitly the local Fourier transform of monomial representations satisfying a certain ramification condition, and deduce Laumon's formula relating the ε-factor to the determinant of the local Fourier transform under the same condition.

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