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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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