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Karol Kozioł
A Classification of the Irreducible mod-$p$ Representations of $\textnormal{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$
(Une classification des représentations irréductibles modulo $p$ de $\textnormal{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$)
Annales de l'institut Fourier, 66 no. 4 (2016), p. 1545-1582, doi: 10.5802/aif.3043
Article PDF
Class. Math.: 22E50, 11F80, 11F85
Keywords: Supersingular representations, unitary group, mod-$p$ representations
[1] Ramla Abdellatif, Autour des répresentations modulo $p$ des groupes réductifs $p$-adiques de rang 1, Ph. D. Thesis, Université Paris-Sud XI (France), 2011
[2] Ramla Abdellatif, “Induction parabolique modulo $p$ pour les groupes $p$-adiques quasi-déployés de rang 1”, Preprint, available at http://perso.ens-lyon.fr/ramla.abdellatif/Articles/rang1.pdf, 2013
[3] Ramla Abdellatif, “Classification des représentations modulo $p$ de ${\rm SL}(2,F)$”, Bull. Soc. Math. France 142 (2014) no. 3, p. 537-589 MR 3295722
[4] Noriyuki Abe, Guy Henniart, Florian Herzig & Marie-France Vignéras, “A classification of irreducible admissible mod $p$ representations of $p$-adic reductive groups”, Preprint, available at http://www.math.toronto.edu/~herzig/classification-apr9.pdf, 2014
[5] L. Barthel & R. Livné, “Irreducible modular representations of ${\rm GL}_2$ of a local field”, Duke Math. J. 75 (1994) no. 2, p. 261-292 Article | MR 1290194 | Zbl 0826.22019
[6] L. Barthel & R. Livné, “Modular representations of ${\rm GL}_2$ of a local field: the ordinary, unramified case”, J. Number Theory 55 (1995) no. 1, p. 1-27 Article | MR 1361556 | Zbl 0841.11026
[7] Joël Bellaïche & Gaëtan Chenevier, “Families of Galois representations and Selmer groups”, Astérisque (2009) no. 324 Zbl 1192.11035
[8] Laurent Berger, “Central characters for smooth irreducible modular representations of ${\rm GL}_2({\bf Q}_p)$”, Rend. Semin. Mat. Univ. Padova 128 (2012), p. 1-6 (2013) Numdam | MR 3076828 | Zbl 1266.22018
[9] A. Borel, 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, p. 27–61 MR 546608 | Zbl 0412.10017
[10] Christophe Breuil, “Sur quelques représentations modulaires et $p$-adiques de ${\rm GL}_2(\mathbf{Q}_p)$. I”, Compositio Math. 138 (2003) no. 2, p. 165-188 Article | MR 2018825 | Zbl 1044.11041
[11] Christophe Breuil, “Sur quelques représentations modulaires et $p$-adiques de ${\rm GL}_2(\mathbf{Q}_p)$. II”, J. Inst. Math. Jussieu 2 (2003) no. 1, p. 23-58 Article | MR 1955206 | Zbl 1165.11319
[12] Christophe Breuil, “Invariant $\mathscr{L}$ et série spéciale $p$-adique”, Ann. Sci. École Norm. Sup. (4) 37 (2004) no. 4, p. 559-610 Article | MR 2097893 | Zbl 1166.11331
[13] Christophe Breuil, “Representations of Galois and $\textnormal{GL}_2$ in characteristic $p$”, Notes for a course given at Columbia University, available at http://www.math.u-psud.fr/~breuil/PUBLICATIONS/New-York.pdf, 2007
[14] Kevin Buzzard & Toby Gee, “The conjectural connections between automorphic representations and Galois representations”, Proceedings of the LMS Durham Symposium 2011, 2013
[15] Jin-Hwan Cho, Mikiya Masuda & Dong Youp Suh, “Extending representations of $H$ to $G$ with discrete $G/H$”, J. Korean Math. Soc. 43 (2006) no. 1, p. 29-43 Article | MR 2190693 | Zbl 1083.22002
[16] Pierre Colmez, “Représentations de ${\rm GL}_2(\mathbf{Q}_p)$ et $(\phi ,\Gamma )$-modules”, Astérisque (2010) no. 330, p. 281-509 MR 2642409 | Zbl 1218.11107
[17] Matthew Emerton, “Local-global compatibility in the $p$-adic Langlands programme for $\textrm{GL}_{2/\mathbb{Q}}$”, Preprint, available at http://www.math.uchicago.edu/~emerton/pdffiles/lg.pdf, 2010 MR 2251474
[18] Mark Kisin, “The Fontaine-Mazur conjecture for ${\rm GL}_2$”, J. Amer. Math. Soc. 22 (2009) no. 3, p. 641-690 Article | MR 2505297 | Zbl 1251.11045
[19] Mark Kisin, “Deformations of $G_{\mathbb{Q}_p}$ and ${\rm GL}_2(\mathbb{Q}_p)$ representations”, Astérisque (2010) no. 330, p. 511-528 MR 2642410 | Zbl 1233.11126
[20] Karol Kozioł & Peng Xu, “Hecke modules and supersingular representations of ${\rm U}(2,1)$”, Represent. Theory 19 (2015), p. 56-93 Article | MR 3321473
[21] J.-P. Labesse & R. P. Langlands, “$L$-indistinguishability for ${\rm SL}(2)$”, Canad. J. Math. 31 (1979) no. 4, p. 726-785 Article | MR 540902 | Zbl 0421.12014
[22] Alberto Mínguez, Unramified representations of unitary groups, On the stabilization of the trace formula, Stab. Trace Formula Shimura Var. Arith. Appl. 1, Int. Press, Somerville, MA, 2011, p. 389–410 MR 2856377
[23] Jürgen Neukirch, Algebraic number theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 322, Springer-Verlag, Berlin, 1999, Translated from the 1992 German original and with a note by Norbert Schappacher, With a foreword by G. Harder Article | MR 1697859 | Zbl 0956.11021
[24] Vytautas Paškūnas, “The image of Colmez’s Montreal functor”, Publ. Math. Inst. Hautes Études Sci. 118 (2013), p. 1-191 Article | MR 3150248 | Zbl 1297.22021
[25] Jonathan D. Rogawski, Automorphic representations of unitary groups in three variables, Annals of Mathematics Studies 123, Princeton University Press, Princeton, NJ, 1990 MR 1081540 | Zbl 0724.11031
[26] Jonathan D. Rogawski, Analytic expression for the number of points mod $p$, The zeta functions of Picard modular surfaces, Univ. Montréal, Montreal, QC, 1992, p. 65–109 MR 1155227 | Zbl 0821.14015
[27] Jean-Pierre Serre, Corps locaux, Hermann, Paris, 1968, Deuxième édition, Publications de l’Université de Nancago, No. VIII MR 354618 | Zbl 0137.02601
[28] Jean-Pierre Serre, 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, p. 193–268 MR 450201 | Zbl 0366.10022
[29] Marie-France Vignéras, Représentations $l$-modulaires d’un groupe réductif $p$-adique avec $l\ne p$, Progress in Mathematics 137, Birkhäuser Boston, Inc., Boston, MA, 1996 Zbl 0859.22001
[30] Marie-France Vignéras, À propos d’une conjecture de Langlands modulaire, Finite reductive groups (Luminy, 1994), Progr. Math. 141, Birkhäuser Boston, Boston, MA, 1997, p. 415–452 MR 1429882 | Zbl 0893.11048
[31] Marie-France Vignéras, “Pro-$p$-Iwahori Hecke ring and supersingular $\overline{\mathbf{F}}_p$-representations”, Math. Ann. 331 (2005) no. 3, p. 523-556 Article | MR 2122539 | Zbl 1107.22011
[32] Marie-France Vignéras, “The right adjoint of the parabolic induction”, Preprint, available at http://webusers.imj-prg.fr/~marie-france.vigneras/ordinaryfunctor2013oct.pdf, 2013 MR 584073
[33] Marie-France Vignéras, “The pro-$p$-Iwahori Hecke algebra of a reductive $p$-adic group I”, To appear in Compos. Math., Preprint available at http://webusers.imj-prg.fr/~marie-france.vigneras/rv2013-18-07.pdf, 2014 MR 3484112
Karol Kozioł
A Classification of the Irreducible mod-$p$ Representations of $\textnormal{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$
(Une classification des représentations irréductibles modulo $p$ de $\textnormal{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$)
Annales de l'institut Fourier, 66 no. 4 (2016), p. 1545-1582, doi: 10.5802/aif.3043
Article PDF
Class. Math.: 22E50, 11F80, 11F85
Keywords: Supersingular representations, unitary group, mod-$p$ representations
Résumé - Abstract
Let $p$ be a prime number. We classify all smooth irreducible mod-$p$ representations of the unramified unitary group $\textnormal{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$ in two variables. We then investigate Langlands parameters in characteristic $p$ associated to $\textnormal{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$, and propose a correspondence between certain equivalence classes of Langlands parameters and certain isomorphism classes of semisimple $L$-packets on $\textnormal{U}(1,1)(\mathbb{Q}_{p^2}/\mathbb{Q}_p)$.
Bibliography
[2] Ramla Abdellatif, “Induction parabolique modulo $p$ pour les groupes $p$-adiques quasi-déployés de rang 1”, Preprint, available at http://perso.ens-lyon.fr/ramla.abdellatif/Articles/rang1.pdf, 2013
[3] Ramla Abdellatif, “Classification des représentations modulo $p$ de ${\rm SL}(2,F)$”, Bull. Soc. Math. France 142 (2014) no. 3, p. 537-589 MR 3295722
[4] Noriyuki Abe, Guy Henniart, Florian Herzig & Marie-France Vignéras, “A classification of irreducible admissible mod $p$ representations of $p$-adic reductive groups”, Preprint, available at http://www.math.toronto.edu/~herzig/classification-apr9.pdf, 2014
[5] L. Barthel & R. Livné, “Irreducible modular representations of ${\rm GL}_2$ of a local field”, Duke Math. J. 75 (1994) no. 2, p. 261-292 Article | MR 1290194 | Zbl 0826.22019
[6] L. Barthel & R. Livné, “Modular representations of ${\rm GL}_2$ of a local field: the ordinary, unramified case”, J. Number Theory 55 (1995) no. 1, p. 1-27 Article | MR 1361556 | Zbl 0841.11026
[7] Joël Bellaïche & Gaëtan Chenevier, “Families of Galois representations and Selmer groups”, Astérisque (2009) no. 324 Zbl 1192.11035
[8] Laurent Berger, “Central characters for smooth irreducible modular representations of ${\rm GL}_2({\bf Q}_p)$”, Rend. Semin. Mat. Univ. Padova 128 (2012), p. 1-6 (2013) Numdam | MR 3076828 | Zbl 1266.22018
[9] A. Borel, 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, p. 27–61 MR 546608 | Zbl 0412.10017
[10] Christophe Breuil, “Sur quelques représentations modulaires et $p$-adiques de ${\rm GL}_2(\mathbf{Q}_p)$. I”, Compositio Math. 138 (2003) no. 2, p. 165-188 Article | MR 2018825 | Zbl 1044.11041
[11] Christophe Breuil, “Sur quelques représentations modulaires et $p$-adiques de ${\rm GL}_2(\mathbf{Q}_p)$. II”, J. Inst. Math. Jussieu 2 (2003) no. 1, p. 23-58 Article | MR 1955206 | Zbl 1165.11319
[12] Christophe Breuil, “Invariant $\mathscr{L}$ et série spéciale $p$-adique”, Ann. Sci. École Norm. Sup. (4) 37 (2004) no. 4, p. 559-610 Article | MR 2097893 | Zbl 1166.11331
[13] Christophe Breuil, “Representations of Galois and $\textnormal{GL}_2$ in characteristic $p$”, Notes for a course given at Columbia University, available at http://www.math.u-psud.fr/~breuil/PUBLICATIONS/New-York.pdf, 2007
[14] Kevin Buzzard & Toby Gee, “The conjectural connections between automorphic representations and Galois representations”, Proceedings of the LMS Durham Symposium 2011, 2013
[15] Jin-Hwan Cho, Mikiya Masuda & Dong Youp Suh, “Extending representations of $H$ to $G$ with discrete $G/H$”, J. Korean Math. Soc. 43 (2006) no. 1, p. 29-43 Article | MR 2190693 | Zbl 1083.22002
[16] Pierre Colmez, “Représentations de ${\rm GL}_2(\mathbf{Q}_p)$ et $(\phi ,\Gamma )$-modules”, Astérisque (2010) no. 330, p. 281-509 MR 2642409 | Zbl 1218.11107
[17] Matthew Emerton, “Local-global compatibility in the $p$-adic Langlands programme for $\textrm{GL}_{2/\mathbb{Q}}$”, Preprint, available at http://www.math.uchicago.edu/~emerton/pdffiles/lg.pdf, 2010 MR 2251474
[18] Mark Kisin, “The Fontaine-Mazur conjecture for ${\rm GL}_2$”, J. Amer. Math. Soc. 22 (2009) no. 3, p. 641-690 Article | MR 2505297 | Zbl 1251.11045
[19] Mark Kisin, “Deformations of $G_{\mathbb{Q}_p}$ and ${\rm GL}_2(\mathbb{Q}_p)$ representations”, Astérisque (2010) no. 330, p. 511-528 MR 2642410 | Zbl 1233.11126
[20] Karol Kozioł & Peng Xu, “Hecke modules and supersingular representations of ${\rm U}(2,1)$”, Represent. Theory 19 (2015), p. 56-93 Article | MR 3321473
[21] J.-P. Labesse & R. P. Langlands, “$L$-indistinguishability for ${\rm SL}(2)$”, Canad. J. Math. 31 (1979) no. 4, p. 726-785 Article | MR 540902 | Zbl 0421.12014
[22] Alberto Mínguez, Unramified representations of unitary groups, On the stabilization of the trace formula, Stab. Trace Formula Shimura Var. Arith. Appl. 1, Int. Press, Somerville, MA, 2011, p. 389–410 MR 2856377
[23] Jürgen Neukirch, Algebraic number theory, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 322, Springer-Verlag, Berlin, 1999, Translated from the 1992 German original and with a note by Norbert Schappacher, With a foreword by G. Harder Article | MR 1697859 | Zbl 0956.11021
[24] Vytautas Paškūnas, “The image of Colmez’s Montreal functor”, Publ. Math. Inst. Hautes Études Sci. 118 (2013), p. 1-191 Article | MR 3150248 | Zbl 1297.22021
[25] Jonathan D. Rogawski, Automorphic representations of unitary groups in three variables, Annals of Mathematics Studies 123, Princeton University Press, Princeton, NJ, 1990 MR 1081540 | Zbl 0724.11031
[26] Jonathan D. Rogawski, Analytic expression for the number of points mod $p$, The zeta functions of Picard modular surfaces, Univ. Montréal, Montreal, QC, 1992, p. 65–109 MR 1155227 | Zbl 0821.14015
[27] Jean-Pierre Serre, Corps locaux, Hermann, Paris, 1968, Deuxième édition, Publications de l’Université de Nancago, No. VIII MR 354618 | Zbl 0137.02601
[28] Jean-Pierre Serre, 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, p. 193–268 MR 450201 | Zbl 0366.10022
[29] Marie-France Vignéras, Représentations $l$-modulaires d’un groupe réductif $p$-adique avec $l\ne p$, Progress in Mathematics 137, Birkhäuser Boston, Inc., Boston, MA, 1996 Zbl 0859.22001
[30] Marie-France Vignéras, À propos d’une conjecture de Langlands modulaire, Finite reductive groups (Luminy, 1994), Progr. Math. 141, Birkhäuser Boston, Boston, MA, 1997, p. 415–452 MR 1429882 | Zbl 0893.11048
[31] Marie-France Vignéras, “Pro-$p$-Iwahori Hecke ring and supersingular $\overline{\mathbf{F}}_p$-representations”, Math. Ann. 331 (2005) no. 3, p. 523-556 Article | MR 2122539 | Zbl 1107.22011
[32] Marie-France Vignéras, “The right adjoint of the parabolic induction”, Preprint, available at http://webusers.imj-prg.fr/~marie-france.vigneras/ordinaryfunctor2013oct.pdf, 2013 MR 584073
[33] Marie-France Vignéras, “The pro-$p$-Iwahori Hecke algebra of a reductive $p$-adic group I”, To appear in Compos. Math., Preprint available at http://webusers.imj-prg.fr/~marie-france.vigneras/rv2013-18-07.pdf, 2014 MR 3484112
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