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Representations of \(\text{GL}_2(\mathbb Q_p)\) and \((\varphi,\Gamma)\)-modules. (Représentations de \(\text{GL}_2(\mathbb Q_p)\) et \((\varphi,\Gamma)\)-modules.) (English) Zbl 1218.11107
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, 281-509 (2010).
Summary: Let \(L\) be a finite extension of \(\mathbb Q_p\). We construct a (\(p\)-adic local Langlands) correspondence attaching to any irreducible, 2-dimensional, \(L\)-representation \(V\) of \(\mathcal G_{\mathbb Q_p}\), a unitary, admissible, irreducible \(L\)-representation \(\Pi(V)\) of \(\mathrm{GL}_2(\mathbb Q_p)\). We identify the locally analytic and locally algebraic vectors of \(\Pi(V)\), which allows us to show that this correspondence encodes the classical local Langlands correspondence (for \(\mathrm{GL}_2(\mathbb Q_p)\)).
For the entire collection see [Zbl 1192.11001].

11S37 Langlands-Weil conjectures, nonabelian class field theory
11F80 Galois representations
22E50 Representations of Lie and linear algebraic groups over local fields
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