Hostname: page-component-848d4c4894-ttngx Total loading time: 0 Render date: 2024-06-08T00:47:42.810Z Has data issue: false hasContentIssue false
  • English
  • Français

Locally analytic vectors of some crystabelian representations of GL2(ℚp)

Part of: Discontinuous groups and automorphic forms Lie groups

Published online by Cambridge University Press:  20 December 2011

Ruochuan Liu
Ruochuan Liu*
Affiliation:
University of Michigan, Ann Arbor, Michigan, USA (email: ruochuan@umich.edu)
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

For V a two-dimensional p-adic representation of Gℚp, we denote by B(V ) the admissible unitary representation of GL2(ℚp) attached to V under the p-adic local Langlands correspondence of GL2(ℚp) initiated by Breuil. In this paper, building on the works of Berger–Breuil and Colmez, we determine the locally analytic vectors B(V )an of B (V ) when V is irreducible, crystabelian and Frobenius semisimple with distinct Hodge–Tate weights; this proves a conjecture of Breuil. Using this result, we verify Emerton’s conjecture that dim Ref η⊗ψ (V )=dim Exp η∣⋅∣⊗xψ (B (V )an ⊗(x∣⋅∣∘det )) for those V which are irreducible, crystabelian and Frobenius semisimple.

Keywords

p-adic local Langlands correspondence(φ,Γ)-modulesGalois representations

MSC classification

Secondary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields 11F80: Galois representations 11F85: $p$-adic theory, local fields 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 148 , Issue 1 , January 2012 , pp. 28 - 64
Copyright
Copyright © Foundation Compositio Mathematica 2011

References

[Ber02]Berger, L., Représentations p-adiques et équations différentielles, Invent. Math. 148 (2002), 219–284.CrossRefGoogle Scholar
[BB10]Berger, L. and Breuil, C., Sur quelques représentations potentiellement cristallines de GL2(ℚp), Astérisque 330 (2010), 155–211.Google Scholar
[Bre04]Breuil, C., Invariant 𝔏 et série spéciale p-adique, Ann. Sci. Éc. Norm. Supér. (4) 37 (2004), 559–610.CrossRefGoogle Scholar
[Bum98]Bump, D., Automorphic forms and representations, Cambridge Studies in Advanced Mathematics, vol. 55 (Cambridge University Press, Cambridge, 1998).Google Scholar
[CC98]Cherbonnier, F. and Colmez, P., Représentations p-adiques surconvergentes, Invent. Math. 133 (1998), 581–611.CrossRefGoogle Scholar
[Col05]Colmez, P., Série principale unitaire pour GL2(ℚp) et représentations triangulines de dimension 2, Preprint (2005), unpublished.Google Scholar
[Col08]Colmez, P., Représentations triangulines de dimension 2, Astérisque 319 (2008), 213–258.Google Scholar
[Col10a]Colmez, P., (φ,Γ)-modules et représentation du mirabolique de GL2(ℚp), Astérisque 330 (2010), 61–153.Google Scholar
[Col10b]Colmez, P., La série principale unitaire de GL2(ℚp), Astérisque 330 (2010), 213–262.Google Scholar
[Col10c]Colmez, P., Fonctions d’une variable p-adic, Astérisque 330 (2010), 13–59.Google Scholar
[Col10d]Colmez, P., Représentations de GL2(ℚp) et (φ,Γ)-modules, Astérisque 330 (2010), 281–509.Google Scholar
[Del71]Deligne, P., Formes modulaires et représentations l-adiques, in Sém. Bourbaki 1968/1969, exp. 343, Springer Lecture Notes, vol. 179 (Springer, 1971), 139–172.Google Scholar
[Eme06a]Emerton, M., A local–global compatibility conjecture in the p-adic Langlands programme for GL2/ℚ, Pure Appl. Math. Q. 2 (2006), 279–393.CrossRefGoogle Scholar
[Eme06b]Emerton, M., Jacquet modules of locally analytic representations of p-adic reductive groups I. Construction and first properties, Ann. Sci. Éc. Norm. Supér. 39 (2006), 353–392.Google Scholar
[Fon90]Fontaine, J.-M., Représentations p-adiques des corps locaux I, in The Grothendieck Festschrift II, Progress in Mathematics, vol. 87 (Birkhäuser, Basel, 1990), 249–309.Google Scholar
[Fon94]Fontaine, J.-M., Représentation l-adiques potentiellement semi-stables. Périodes p-adiques, Astérisque 223 (1994), 321–347.Google Scholar
[Ked08]Kedlaya, K., Slope filtrations for relative Frobenius, Astérisque 319 (2008), 259–301.Google Scholar
[Pas09]Paskunas, V., On some crystalline representations of GL2(ℚp), Algebra Number Theory 3 (2009), 411–421.CrossRefGoogle Scholar
[Per01]Perrin-Riou, B., Théorie d’Iwasawa des représentations p-adiques semi-stables, Mém. Soc. Math. Fr. (N.S.) 84 (2001).Google Scholar
[ST01]Schneider, P. and Teitelbaum, J., U(𝔤)-finite locally analytic representations, Represent. Theory 5 (2001), 111–128.CrossRefGoogle Scholar
[ST02a]Schneider, P. and Teitelbaum, J., Locally analytic distributions and p-adic representation theory, with applications to GL2, J. Amer. Math. Soc. 15 (2002), 443–468.CrossRefGoogle Scholar
[ST02b]Schneider, P. and Teitelbaum, J., Banach space representations and Iwasawa theory, Israel J. Math. 127 (2002), 359–380.CrossRefGoogle Scholar
[ST03]Schneider, P. and Teitelbaum, J., Algebras of p-adic distributions and admissible representations, Invent. Math. 153 (2003), 145–196.CrossRefGoogle Scholar