## Overconvergence of $$p$$-adic representations: the relative case. (Surconvergence des représentations $$p$$-adiques: le cas relatif.)(French. English summary)Zbl 1168.11018

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, 39-116 (2008).
Let $$K$$ be a local field containing $${\mathbb Q}_p$$. By a result of Cherbonnier-Colmez, the category of $$p$$-adic representations $$V$$ of $$\text{Gal}(\overline{K}/K)$$ is equivalent to the category of étale $$(\phi,\Gamma)$$-modules over the Robba ring associated to $$K$$: this last category is, in principle, much more accessible than the first one. Moreover, from objects in this last category, ‘geometric’ invariants of $$p$$-adic representations $$V$$ of $$\text{Gal}(\overline{K}/K)$$ can be read off much easier than from the more classical étale $$(\phi,\Gamma)$$-modules over Fontaine’s field $${\mathcal E}$$.
The purpose of this article is to generalize this equivalence of categories: now the category of $$p$$-adic representations of $$\text{Gal}(\overline{R}\,[p^{-1}]/R\,[p^{-1}])$$ is shown to be equivalent to certain objects generalizing the above étale $$(\phi,\Gamma)$$-modules over the Robba ring. Here the rings $$R$$ (respectively extensions $$\overline{R}$$ of $$R$$) describe, very roughly, small affine/affinoid pieces in smooth $${\mathcal O}_K$$-varieties (resp. generically étale coverings of such). (The authors refer to this setting as the ‘relative’ version of the ‘absolute’ one described above.)
One of the crucial points is to carry out Sen-type ‘decompletion’ arguments, as formalized by Colmez (i.e. the verification of ‘overconvergence’ of certain Laurent series).
For the entire collection see [Zbl 1156.14002].

### MSC:

 11F80 Galois representations 11F85 $$p$$-adic theory, local fields 11S25 Galois cohomology 11S15 Ramification and extension theory 11S20 Galois theory 13K05 Witt vectors and related rings (MSC2000) 14E22 Ramification problems in algebraic geometry 14F30 $$p$$-adic cohomology, crystalline cohomology