On non-admissible irreducible modulo $p$ representations of ${\mathrm{GL}}_2\left({\mathbb{Q}}_{p^2}\right)$

Ghate, Eknath; Sheth, Mihir

Comptes Rendus. Mathématique, Tome 358 (2020) no. 5, pp. 627-632

We use a Diamond diagram attached to a 2-dimensional reducible split mod $p$ Galois representation of $\mathrm{Gal}(\overline{{\mathbb{Q}}_p}/{\mathbb{Q}}_{p^2})$ to construct a non-admissible smooth irreducible mod $p$ representation of ${\mathrm{GL}}_2\left({\mathbb{Q}}_{p^2}\right)$ following the approach of Daniel Le.

Nous utilisons un diagramme de Diamond attaché à une représentation galoisienne mod $p$ semi-simple réductible de dimension 2 de $\mathrm{Gal}(\overline{{\mathbb{Q}}_p}/{\mathbb{Q}}_{p^2})$ pour construire une représentation mod $p$ non-admissible irréductible lisse de ${\mathrm{GL}}_2\left({\mathbb{Q}}_{p^2}\right)$ en suivant l’approche de Daniel Le.

Classification : 22E50, 11S37

[1] Abe, Noriyuki; Henniart, Guy; Herzig, Florian; Vigneras, Marie-France Questions on mod $p$ representations of reductive $p$-adic groups (2017) (https://arxiv.org/abs/1703.02063)

[2] Berger, Laurent Central characters for smooth irreducible modular representations of ${\mathrm{GL}}_2\left({\mathbb{Q}}_p\right)$, Rend. Semin. Mat. Univ. Padova, Volume 128 (2012), pp. 1-6 | MR | DOI | Zbl

[3] Bernstein, Joseph All reductive p-adic groups are of type I, Funkts. Anal. Prilozh., Volume 8 (1974) no. 2, pp. 3-6 | MR

[4] Breuil, Christophe Representations of Galois and of ${\mathrm{GL}}_2$ in characteristic $p$ (2007) (Lecture notes of a graduate course at Columbia University)

[5] Breuil, Christophe; Paškūnas, Vytautas Towards a modulo $p$ Langlands correspondence for ${\mathrm{GL}}_2$, Memoirs of the American Mathematical Society, 216, American Mathematical Society, 2012 | MR | Zbl

[6] Harish-Chandra Harmonic analysis on reductive p-adic groups. Notes by G. van Dijk, Lecture Notes in Mathematics, 162, Springer, 1970 | Zbl

[7] Henniart, Guy; Vignéras, Marie-France Representations of a $p$-adic group in characteristic $p$, Representations of reductive groups (Proceedings of Symposia in Pure Mathematics), Volume 101, American Mathematical Society, 2019, pp. 171-210 | DOI | MR

[8] Jacquet, Hervé Sur les représentations des groupes réductifs $p$-adiques, C. R. Math. Acad. Sci. Paris, Volume 280 (1975), pp. 1271-1272 | Zbl

[9] Le, Daniel On some non-admissible smooth representations of ${\mathrm{GL}}_2$, Math. Res. Lett., Volume 26 (2019) no. 6, pp. 1747-1758 | MR

[10] Paškūnas, Vytautas Coefficient systems and supersingular representations of ${\mathrm{GL}}_2\left(F\right)$, Mémoires de la Société Mathématique de France, 99, Société Mathématique de France, 2004 | Zbl | MR | Numdam

[11] Schraen, Benjamin Sur la présentation des représentations supersingulières de ${\mathrm{GL}}_2\left(F\right)$, J. Reine Angew. Math., Volume 704 (2015), pp. 187-208 | MR | Zbl

[12] Vignéras, Marie-France Représentations $l$-modulaires d’un groupe réductif $p$-adique avec $l\ne p$, Progress in Mathematics, 137, Birkhäuser, 1996 | Zbl