The unitary principal series of \(\text{GL}_2 (\mathbb Q_p)\). (La série principale unitaire de \(\text{GL}_2 (\mathbb Q_p)\).) (French. English summary) Zbl 1242.11095

Colmez, Pierre

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, 213-262 (2010).

This paper is devoted to the \(p\)-adic local Langlands correspondence for the unitary principal series of \(\text{GL}_2 (\mathbb Q_p)\). The author uses Fontaine’s theory of \((\varphi,\Gamma)\)-modules to establish, in the semistable and non geometric cases, a direct link between the (dual of) representation of \(\text{GL}_2 (\mathbb Q_p)\) and the \((\varphi,\Gamma)\)-module attached to the representation of \(\text{Gal}(\overline{\mathbb Q}_p/\mathbb Q_p)\).
For the entire collection see [Zbl 1192.11001].

Reviewer: L. N. Vaserstein (University Park)

MSC:

11S37

Langlands-Weil conjectures, nonabelian class field theory

Keywords:

local Langlands correspondence