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].


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