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Two-dimensional trianguline representations. (Représentations triangulines de dimension 2.) (French. English summary) Zbl 1168.11022
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, 213-258 (2008).
In this paper the notion of trianguline $$p$$-adic representations of $$\text{Gal}(\overline{{\mathbb Q}_p}/{\mathbb Q}_p)$$ is defined as follows. A $$(\phi,\Gamma)$$-module over the Robba ring $${\mathcal R}$$ over a coefficient field $$L$$ is trianguline if it is a successive extension of $$(\phi,\Gamma)$$-modules of rank 1. As the category of $$p$$-adic $$L$$-representations of $$\text{Gal}(\overline{{\mathbb Q}_p}/{\mathbb Q}_p)$$ is equivalent to the category of $$(\phi,\Gamma)$$-modules over $${\mathcal R}$$, one gets the notion of a trianguline $$p$$-adic representation of $$\text{Gal}(\overline{{\mathbb Q}_p}/{\mathbb Q}_p)$$.
The two-dimensional trianguline $$p$$-adic representations of $$\text{Gal}(\overline{{\mathbb Q}_p}/{\mathbb Q}_p)$$ are studied in detail: they are the local analogues of Galois representations attached to finite slope modular forms.
For the entire collection see [Zbl 1156.14002].

##### MSC:
 11F80 Galois representations 11F85 $$p$$-adic theory, local fields 11S37 Langlands-Weil conjectures, nonabelian class field theory 11S25 Galois cohomology 11S15 Ramification and extension theory