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Special values of parameters in Diamond diagrams. (Valeurs spéciales de paramètres de diagrammes de Diamond.) (French. English summary) Zbl 1355.22004
Although the Langlands correspondence modulo a prime $$p$$ for $$\text{GL}_2 (\mathbb Q_p)$$ is well understood, the case for $$\text{GL}_2 (L)$$ with $$L \neq \mathbb Q_p$$ remains largely mysterious. The main obstacle in this case is the existence of many more representations of $$\text{GL}_2 (L)$$ than 2-dimensional representations of $$\text{Gal} (\overline{\mathbb Q}_p/L)$$. Let $$L$$ be a finite unramified extension of $$\mathbb Q_p$$, and let $$\overline{\rho}: \text{Gal} (\overline{\mathbb Q}_p/L) \to \text{GL}_2 (\overline{\mathbb F}_p)$$ be a reducible continuous generic representation. In this paper, the author follows the strategy described in the paper of C. Breuil and F. Diamond [Ann. Sci. Éc. Norm. Supér. (4) 47, No. 5, 905–974 (2014; Zbl 1309.11046)] to prove how the extension type of $$\overline{\rho}$$ is related to certain parameters that appear in the Diamond diagrams associated to $$\overline{\rho}$$.
##### MSC:
 22E50 Representations of Lie and linear algebraic groups over local fields 11F85 $$p$$-adic theory, local fields 11F70 Representation-theoretic methods; automorphic representations over local and global fields
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