A Classification of the Irreducible mod-$p$ Representations of $U(1,1)({\mathbb{Q}}_{p^2}/{\mathbb{Q}}_p)$

Kozioł, Karol

Annales de l'Institut Fourier, Tome 66 (2016) no. 4, pp. 1545-1582.

Soit $p$ un nombre premier. Nous classifions les représentations lisses irréductibles modulo $p$ du groupe unitaire non-ramifié $U(1,1)({\mathbb{Q}}_{p^2}/{\mathbb{Q}}_p)$ en deux variables. Ensuite, nous étudions les paramètres de Langlands en caractéristique $p$ associés à $U(1,1)({\mathbb{Q}}_{p^2}/{\mathbb{Q}}_p)$ et proposons une correspondance entre certaines classes d’équivalence de paramètres de Langlands et certaines classes d’isomorphisme de $L$-paquets semi-simples de $U(1,1)({\mathbb{Q}}_{p^2}/{\mathbb{Q}}_p)$.

Let $p$ be a prime number. We classify all smooth irreducible mod-$p$ representations of the unramified unitary group $U(1,1)({\mathbb{Q}}_{p^2}/{\mathbb{Q}}_p)$ in two variables. We then investigate Langlands parameters in characteristic $p$ associated to $U(1,1)({\mathbb{Q}}_{p^2}/{\mathbb{Q}}_p)$, and propose a correspondence between certain equivalence classes of Langlands parameters and certain isomorphism classes of semisimple $L$-packets on $U(1,1)({\mathbb{Q}}_{p^2}/{\mathbb{Q}}_p)$.

Classification : 22E50, 11F80, 11F85
Keywords: Supersingular representations, unitary group, mod-$p$ representations
Mot clés : Représentations supersingulières, groupe unitaire, représentations modulo $p$

[1] Abdellatif, Ramla Autour des répresentations modulo $p$ des groupes réductifs $p$-adiques de rang 1, Université Paris-Sud XI (France) (2011) (Ph. D. Thesis)

[2] Abdellatif, Ramla Induction parabolique modulo $p$ pour les groupes $p$-adiques quasi-déployés de rang 1 (2013) (Preprint, available at http://perso.ens-lyon.fr/ramla.abdellatif/Articles/rang1.pdf)

[3] Abdellatif, Ramla Classification des représentations modulo $p$ de $\mathrm{SL}(2,F)$, Bull. Soc. Math. France, Volume 142 (2014) no. 3, pp. 537-589

[4] Abe, Noriyuki; Henniart, Guy; Herzig, Florian; Vignéras, Marie-France A classification of irreducible admissible mod $p$ representations of $p$-adic reductive groups (2014) (Preprint, available at http://www.math.toronto.edu/~herzig/classification-apr9.pdf)

[5] Barthel, L.; Livné, R. Irreducible modular representations of ${\mathrm{GL}}_2$ of a local field, Duke Math. J., Volume 75 (1994) no. 2, pp. 261-292 | DOI

[6] Barthel, L.; Livné, R. Modular representations of ${\mathrm{GL}}_2$ of a local field: the ordinary, unramified case, J. Number Theory, Volume 55 (1995) no. 1, pp. 1-27 | DOI

[7] Bellaïche, Joël; Chenevier, Gaëtan Families of Galois representations and Selmer groups, Astérisque (2009) no. 324, xii+314 pages

[8] Berger, Laurent Central characters for smooth irreducible modular representations of ${\mathrm{GL}}_2\left({\mathbf{Q}}_p\right)$, Rend. Semin. Mat. Univ. Padova, Volume 128 (2012), p. 1-6 (2013) | DOI

[9] Borel, A. Automorphic $L$-functions, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2 (Proc. Sympos. Pure Math., XXXIII), Amer. Math. Soc., Providence, R.I., 1979, pp. 27-61

[10] Breuil, Christophe Sur quelques représentations modulaires et $p$-adiques de ${\mathrm{GL}}_2\left({\mathbf{Q}}_p\right)$. I, Compositio Math., Volume 138 (2003) no. 2, pp. 165-188 | DOI

[11] Breuil, Christophe Sur quelques représentations modulaires et $p$-adiques de ${\mathrm{GL}}_2\left({\mathbf{Q}}_p\right)$. II, J. Inst. Math. Jussieu, Volume 2 (2003) no. 1, pp. 23-58 | DOI

[12] Breuil, Christophe Invariant $ℒ$ et série spéciale $p$-adique, Ann. Sci. École Norm. Sup. (4), Volume 37 (2004) no. 4, pp. 559-610 | DOI

[13] Breuil, Christophe Representations of Galois and $G{L}_2$ in characteristic $p$ (2007) (Notes for a course given at Columbia University, available at http://www.math.u-psud.fr/~breuil/PUBLICATIONS/New-York.pdf)

[14] Buzzard, Kevin; Gee, Toby The conjectural connections between automorphic representations and Galois representations (2013) (Proceedings of the LMS Durham Symposium 2011)

[15] Cho, Jin-Hwan; Masuda, Mikiya; Suh, Dong Youp Extending representations of $H$ to $G$ with discrete $G/H$, J. Korean Math. Soc., Volume 43 (2006) no. 1, pp. 29-43 | DOI

[16] Colmez, Pierre Représentations de ${\mathrm{GL}}_2\left({\mathbf{Q}}_p\right)$ et $(\phi ,\Gamma )$-modules, Astérisque (2010) no. 330, pp. 281-509

[17] Emerton, Matthew Local-global compatibility in the $p$-adic Langlands programme for ${\mathrm{GL}}_{2/\mathbb{Q}}$ (2010) (Preprint, available at http://www.math.uchicago.edu/~emerton/pdffiles/lg.pdf)

[18] Kisin, Mark The Fontaine-Mazur conjecture for ${\mathrm{GL}}_2$, J. Amer. Math. Soc., Volume 22 (2009) no. 3, pp. 641-690 | DOI

[19] Kisin, Mark Deformations of $G_{{\mathbb{Q}}_p}$ and ${\mathrm{GL}}_2\left({\mathbb{Q}}_p\right)$ representations, Astérisque (2010) no. 330, pp. 511-528

[20] Kozioł, Karol; Xu, Peng Hecke modules and supersingular representations of $\mathrm{U}(2,1)$, Represent. Theory, Volume 19 (2015), pp. 56-93 | DOI

[21] Labesse, J.-P.; Langlands, R. P. $L$-indistinguishability for $\mathrm{SL}\left(2\right)$, Canad. J. Math., Volume 31 (1979) no. 4, pp. 726-785 | DOI

[22] Mínguez, Alberto Unramified representations of unitary groups, On the stabilization of the trace formula (Stab. Trace Formula Shimura Var. Arith. Appl.), Volume 1, Int. Press, Somerville, MA, 2011, pp. 389-410

[23] Neukirch, Jürgen Algebraic number theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 322, Springer-Verlag, Berlin, 1999, xviii+571 pages (Translated from the 1992 German original and with a note by Norbert Schappacher, With a foreword by G. Harder) | DOI

[24] Paškūnas, Vytautas The image of Colmez’s Montreal functor, Publ. Math. Inst. Hautes Études Sci., Volume 118 (2013), pp. 1-191 | DOI

[25] Rogawski, Jonathan D. Automorphic representations of unitary groups in three variables, Annals of Mathematics Studies, 123, Princeton University Press, Princeton, NJ, 1990, xii+259 pages

[26] Rogawski, Jonathan D. Analytic expression for the number of points mod $p$, The zeta functions of Picard modular surfaces, Univ. Montréal, Montreal, QC, 1992, pp. 65-109

[27] Serre, Jean-Pierre Corps locaux, Hermann, Paris, 1968, 245 pages (Deuxième édition, Publications de l’Université de Nancago, No. VIII)

[28] Serre, Jean-Pierre Modular forms of weight one and Galois representations, Algebraic number fields: $L$-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), Academic Press, London, 1977, pp. 193-268

[29] Vignéras, Marie-France Représentations $l$-modulaires d’un groupe réductif $p$-adique avec $l\ne p$, Progress in Mathematics, 137, Birkhäuser Boston, Inc., Boston, MA, 1996, xviii and 233 pages

[30] Vignéras, Marie-France À propos d’une conjecture de Langlands modulaire, Finite reductive groups (Luminy, 1994) (Progr. Math.), Volume 141, Birkhäuser Boston, Boston, MA, 1997, pp. 415-452

[31] Vignéras, Marie-France Pro-$p$-Iwahori Hecke ring and supersingular ${\overline{\mathbf{F}}}_p$-representations, Math. Ann., Volume 331 (2005) no. 3, pp. 523-556 | DOI

[32] Vignéras, Marie-France The right adjoint of the parabolic induction (2013) (Preprint, available at http://webusers.imj-prg.fr/~marie-france.vigneras/ordinaryfunctor2013oct.pdf)

[33] Vignéras, Marie-France The pro-$p$-Iwahori Hecke algebra of a reductive $p$-adic group I (2014) (To appear in Compos. Math., Preprint available at http://webusers.imj-prg.fr/~marie-france.vigneras/rv2013-18-07.pdf)