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$$p$$-adic differential equations and filtered $$(\varphi, N)$$-modules. (Équations différentielles $$p$$-adiques et $$(\varphi, N)$$-modules filtrés.) (French. English summary) Zbl 1168.11019
Berger, Laurent (ed.) et al., Représentation $$p$$-adiques de groupes $$p$$-adiques I. Représentations galoisiennes et $$(\varphi, \Gamma)$$-modules. Paris: Société Mathématique de France (ISBN 978-2-85629-256-3/pbk). Astérisque 319, 13-38 (2008).
Let $$K$$ be a local field containing $${\mathbb Q}_p$$. To control the category of $$p$$-adic representations $$V$$ of $$\text{Gal}(\overline{K}/K)$$ various constructions have been investigated which associate to such $$V$$ certain linear algebra objects which, due to their explicit nature, are easier to handle. The goal of this article is to exactly describe the relations between the various categories of linear algebra objects so considered.
First, there is the category of filtered $$(\phi,N,G_K)$$-modules: if $$V$$ is a $$p$$-adic étale cohomology group of an algebraic $$K$$-scheme then the associated filtered $$(\phi,N,G_K)$$-module can be read off from the crystalline/de Rham cohomology, and is called admissible.
Second, there is the category of $$(\varphi,\Gamma_K)$$-modules over the Robba ring, a concept which involves $$p$$-adic differential equations on thin $$p$$-adic annuli: this category is a variation of a linear algebra category capable of understanding all (not necessarily admissible) $$p$$-adic representations of $$\text{Gal}(\overline{K}/K)$$.
For a long time the precise relation between these two kind of objects has been unclear. The main theorem of this text establishes an equivalence between
1. the category of filtered $$(\phi,N,G_K)$$-modules, and
2. the category of $$(\varphi,\Gamma_K)$$-modules over the Robba ring such that the Lie-algebra of $$\Gamma_K$$ acts locally trivial.
As an application a new proof of the result of Colmez-Fontaine is given: weakly admissible filtered $$(\phi,N,G_K)$$-modules (weak admissibility is a completely explicit numerical condition) are admissible.
Finally, now starting with a $$p$$-adic representation $$V$$ of $$\text{Gal}(\overline{K}/K)$$, the linear algebra objects associated to $$V$$ are compared.
For the entire collection see [Zbl 1156.14002].

##### MSC:
 11F80 Galois representations 11F85 $$p$$-adic theory, local fields 12H25 $$p$$-adic differential equations 13K05 Witt vectors and related rings (MSC2000) 14F30 $$p$$-adic cohomology, crystalline cohomology