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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].

11F80 Galois representations
11F85 \(p\)-adic theory, local fields
11S25 Galois cohomology
12H25 \(p\)-adic differential equations
14F30 \(p\)-adic cohomology, crystalline cohomology