Colmez, Pierre 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]. Reviewer: Elmar Große-Klönne (Berlin) Cited in 8 ReviewsCited in 42 Documents 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 Keywords:\(p\)-adic Galois representation; trianguline representation PDFBibTeX XMLCite \textit{P. Colmez}, Astérisque 319, 213--258 (2008; Zbl 1168.11022)