Special values of parameters in Diamond diagrams. (Valeurs spéciales de paramètres de diagrammes de Diamond.) (French. English summary) Zbl 1355.22004

Hu, Yongquan

Bull. Soc. Math. Fr. 144, No. 1, 77-115 (2016).

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 \(\overline{\rho}\).

Reviewer: Min Ho Lee (Cedar Falls)

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

Keywords:

Langlands correspondence; Diamond diagrams; representations of \(p\)-adic groups