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Overconvergent $$p$$-adic representations. (Représentations $$p$$-adiques surconvergentes.) (French) Zbl 0928.11051
Let $$K$$ be a local $$p$$-adic field, $${\mathcal G}_K$$ its absolute Galois group, J.-M. Fontaine’s theory of $$(\varphi,\Gamma)$$-modules associates to each $$p$$-adic representation $$V$$ of $${\mathcal G}_k$$ a module $$D(V)$$ over a certain local field of dimension 2, with natural and commuting actions of the Frobenius $$\varphi$$ and the Galois group $$\Gamma=\text{Gal} (K( \mu_{p^\infty} /K)$$.
The main point is that, starting from $$D(V)$$, one can reconstruct the representation $$V$$, and one hopes to describe in this way all the classical invariants of $$V$$, for instance its Galois cohomology (Laurent Herr’s thesis, 1995) [see L. Herr, Bull. Soc. Math. Fr. 126, 563–600 (1998; Zbl 0967.11050)].
In this paper, the authors show that any $$p$$-adic representation of $${\mathcal G}_K$$ is “overconvergent”. This notion, introduced in the first author’s thesis (1996), is too technical to be recalled here, but it is the essential tool which allows to recover from $$D(V)$$ the more subtle invariants of $$V$$ built upon the rings $$B_{\text{cris}}$$, $$B_{\text{dR}}$$ etc.
The proof, using the action of $$\Gamma$$, is inspired from Shankar Sen’s proof that any $$p$$-adic representation of $${\mathcal G}_K$$ has a generalized Hodge-Tate decomposition [Invent. Math. 62, 89–116 (1980; Zbl 0463.12005)], but of course, it must overcome technical difficulties which appear because the rings involved are more complicated than $$\mathbb C_n$$.
In another paper [“Théorie d’Iwasawa des représentations $$p$$-adiques d’un corps local”, J. Am. Math. Soc. 12, 241–268 (1999; Zbl 0933.11056)] the authors have given applications to Iwasawa theory of local fields and explicit reciprocity laws.

##### MSC:
 11S20 Galois theory 14F30 $$p$$-adic cohomology, crystalline cohomology 11S25 Galois cohomology
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