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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.

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
Full Text: DOI
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