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On some modular representations of the Borel subgroup of GL2(Qp)

Part of: Representation theory of groups Lie groups Discontinuous groups and automorphic forms Graph theory Linear algebraic groups and related topics

Published online by Cambridge University Press:  11 December 2009

Laurent Berger
Laurent Berger*
Affiliation:
Université de Lyon, UMPA ENS Lyon, 46 allée d’Italie, 69007 Lyon, France (email: laurent.berger@umpa.ens-lyon.fr)
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Abstract

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Colmez has given a recipe to associate a smooth modular representation Ω(W) of the Borel subgroup of GL2(Qp) to a -representation W of by using Fontaine’s theory of (φ,Γ)-modules. We compute Ω(W) explicitly and we prove that if W is irreducible and dim (W)=2, then Ω(W) is the restriction to the Borel subgroup of GL2(Qp) of the supersingular representation associated to W by Breuil’s correspondence.

Keywords

p-adic Langlands correspondencesupersingular representations(φ,Γ)-modulesGalois representations

MSC classification

Secondary: 11F80: Galois representations 05C05: Trees 11F33: Congruences for modular and $p$-adic modular forms 11F70: Representation-theoretic methods; automorphic representations over local and global fields 11F85: $p$-adic theory, local fields 20C20: Modular representations and characters 20G25: Linear algebraic groups over local fields and their integers 22E35: Analysis on $p$-adic Lie groups 22E50: Representations of Lie and linear algebraic groups over local fields
Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 1 , January 2010 , pp. 58 - 80
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

[1]Barthel, L. and Livné, R., Irreducible modular representations of GL2 of a local field, Duke Math. J. 75 (1994), 261–292.CrossRefGoogle Scholar
[2]Barthel, L. and Livné, R., Modular representations of GL2 of a local field: the ordinary, unramified case, J. Number Theory 55 (1995), 1–27.CrossRefGoogle Scholar
[3]Berger, L., Représentations modulaires de GL2(Qp) et représentations galoisiennes de dimension 2, Astérisque (2005), to appear.Google Scholar
[4]Breuil, C., Sur quelques représentations modulaires et p-adiques de GL2(Qp). I, Compositio Math. 138 (2003), 165–188.CrossRefGoogle Scholar
[5]Breuil, C., Sur quelques représentations modulaires et p-adiques de GL2(Qp). II, J. Inst. Math. Jussieu 2 (2003), 23–58.CrossRefGoogle Scholar
[6]Colmez, P., (φ,Γ)-modules et représentations du mirabolique de GL2(Qp), Astérisque (2007), to appear.Google Scholar
[7]Colmez, P., Représentations de GL2(Qp) et (φ,Γ)-modules, Astérisque (2008), to appear.Google Scholar
[8]Emerton, M., On a class of coherent rings, with applications to the smooth representation theory of GL2(Qp) in characteristic p, Preprint (2008).Google Scholar
[9]Fontaine, J.-M., Représentations p-adiques des corps locaux. I, in The Grothendieck Festschrift, Vol. II, Progress in Mathematics, vol. 87 (Birkhäuser, Boston, MA, 1990), 249–309.Google Scholar
[10]Fontaine, J.-M., Le corps des périodes p-adiques, Astérisque (1994), 59–111, With an appendix by Pierre Colmez, Périodes p-adiques (Bures-sur-Yvette, 1988).Google Scholar
[11]Fontaine, J.-M. and Wintenberger, J.-P., Le corps des normes de certaines extensions algébriques de corps locaux, C. R. Acad. Sci. Paris Sér. A–B 288 (1979), A367–A370.Google Scholar
[12]Paškūnas, V., On the restriction of representations of GL2(F) to a Borel subgroup, Compositio Math. 143 (2007), 1533–1544.CrossRefGoogle Scholar
[13]Schneider, P. and Vignéras, M.-F., A functor from smooth o-torsion representations to (φ,Γ)-modules, Preprint (2008).Google Scholar
[14]Serre, J.-P., Propriétés galoisiennes des points d’ordre fini des courbes elliptiques, Invent. Math. 15 (1972), 259–331.CrossRefGoogle Scholar
[15]Vignéras, M.-F., Série principale modulo p de groupes réductifs p-adiques, Geom. Funct. Anal. 17 (2008), 2090–2112.CrossRefGoogle Scholar
[16]Wintenberger, J.-P., Le corps des normes de certaines extensions infinies de corps locaux; applications, Ann. Sci. École Norm. Sup. (4) 16 (1983), 59–89.CrossRefGoogle Scholar