## 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

### Keywords:

Galois representations; deformations
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