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Deformations of \(G_{\mathbb Q_p}\) and \(\text{GL}_2(\mathbb Q_p)\) representations. (English) Zbl 1233.11126

Berger, Laurent (ed.) et al., Représentations \(p\)-adiques de groupes \(p\)-adiques II: Représentations de \(\text{GL}_2 (\mathbb Q_p)\) et \((\varphi, \gamma)\)-modules. Paris: Société Mathématique de France (ISBN 978-2-85629-281-5/pbk). Astérisque 330, 511-528 (2010).
Summary: We show that Colmez’s functor from \(\text{GL}_2(\mathbb Q_p)\) representations to \(G_{\mathbb Q_p}\) representation produces essentially all two dimensional representations of \(G_{\mathbb Q_p}\). The method compares the deformation theory for the two kinds of representations: A \(G_{\mathbb Q_p}\) group calculation of Colmez implies that the deformation space for \(\text{GL}_2(\mathbb Q_p)\) representations is closed in that for \(G_{\mathbb Q_p}\)-representations. A local version of the Gouvêa-Mazur “infinite fern” argument shows that this closed subspace is also dense.
For the entire collection see [Zbl 1192.11001].

MSC:

11S37 Langlands-Weil conjectures, nonabelian class field theory
11F80 Galois representations
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