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Semi-stable representations of torsion in the case $$er<p-1$$. (Représentations semi-stable de torsion dans le cas $$er<p-1$$.) (French) Zbl 1134.14013
Let $$K$$ be a finite extension of $$\mathbb{Q}_p$$ and $$G_K$$ its absolute Galois group. Fontaine has defined what it means for a $$p$$-adic representation to be crystalline or semi-stable and to such a representation $$V$$ one can attach a filtered $$(\varphi,N)$$-module $$D(V)$$ which describes it completely. If $$K$$ is unramified and $$V$$ is crystalline and the length of the filtration on $$D(V)$$ is $$<p-1$$ then Fontaine-Laffaille theory gives a bijection between the $$G_K$$-stable lattices of $$V$$ and certain “strongly divisible” lattices of $$D(V)$$. It then makes sense to talk about torsion crystalline representations.
In the article under review, the author extends these results to the case where the index $$e$$ of ramification of $$K$$ is possibly $$>1$$ and $$V$$ is semi-stable, provided that $$er<p-1$$ where $$r$$ is the length of the filtration on $$D(V)$$. The $$G_K$$-stable lattices of $$V$$ are then described by certain “Breuil modules” which are complicated but explicit objects, and it then makes sense to talk about torsion semi-stable representations. The author also determines all isomorphism classes of simple objects in the category he constructs. As a corollary of his constructions, the author can describe the weights of inertia on the reduction modulo $$p$$ of the étale cohomology groups of a proper smooth scheme $$X$$ over $$K$$ with semi-stable reduction.

##### MSC:
 14F30 $$p$$-adic cohomology, crystalline cohomology 14F20 Étale and other Grothendieck topologies and (co)homologies 11F80 Galois representations 11S20 Galois theory 11G25 Varieties over finite and local fields
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##### References:
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