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Families of de Rham representations and $$p$$-adic monodromy. (Familles de représentations de de Rham et monodromie $$p$$-adique.) (French. English summary) Zbl 1168.11020
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, 303-337 (2008).
Let $$K$$ be a local field containing $${\mathbb Q}_p$$. Recall that the category of $$p$$-adic representations of $$\text{Gal}(\overline{K}/K)$$ is equivalent with the corresponding category of $$(\varphi,\Gamma)$$-module over the Robba-ring of $$K$$. The aim of this article is to study $$p$$-adically varying families of $$p$$-adic representations of $$G_K=\text{Gal}(\overline{K}/K)$$, and the behaviour of the associated $$(\varphi,\Gamma)$$-modules.
Let $$S$$ be a $${\mathbb Q}_p$$-Banach algebra whose residue fields at points in the maximal spectrum $${\mathfrak X}$$ are finite extensions of $${\mathbb Q}_p$$. By definition, a family of $$p$$-adic representations of $$G_K=\text{Gal}(\overline{K}/K)$$ is a free $$S$$-module $$V$$ of finite rank endowed with an $$S$$-linear and continuous $$G_K$$-action. Using the methods of Tate-Sen, the authors prove that for such a $$V$$ there exists a $$S\widehat{\otimes}{\mathbf B}_K^{\dagger}$$-module $$D^{\dagger}(V)$$, locally free and stable for $$\varphi$$ and $$\Gamma_K$$, whose fibres at points of $${\mathfrak X}$$ are exactly the usual $$(\varphi,\Gamma)$$-modules assigned to the corresponding fibres of $$V$$.
As applications it is shown that (1) $$p$$-adic representations of $$\text{Gal}(\overline{K}/K)$$ are overconvergent, and (2) in the above setting, the set of points of $${\mathfrak X}$$ where the fibre of $$V$$ is de Rham (or semistable, or crystalline) with Hodge-Tate weights in a fixed interval is an analytic subspace of $${\mathfrak X}$$.
For the entire collection see [Zbl 1156.14002].

##### MSC:
 11F80 Galois representations 11F85 $$p$$-adic theory, local fields 11S25 Galois cohomology 12H25 $$p$$-adic differential equations 14F30 $$p$$-adic cohomology, crystalline cohomology