## 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].

### MSC:

 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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