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Classification of two-dimensional split trianguline representations of p-adic fields

Part of:Discontinuous groups and automorphic formsAlgebraic number theory: local and $p$-adic fields

Published online by Cambridge University Press:  01 July 2009

Kentaro Nakamura
Kentaro Nakamura*
Affiliation:
Department of Mathematical Sciences, University of Tokyo, 153-8914, Tokyo, Japan (email: kentarou@ms.u-tokyo.ac.jp)
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Abstract

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The aim of this article is to classify two-dimensional split trianguline representations of p-adic fields. This is a generalization of a result of Colmez who classified two-dimensional split trianguline representations of $\mathrm {Gal}(\bar {\mathbb {Q}}_p/\mathbb {Q}_p)$ for p≠2 by using (φ,Γ)-modules over a Robba ring. In this article, for any prime p and for any p-adic field K, we classify two-dimensional split trianguline representations of $\mathrm {Gal}(\bar {K}/K)$ using B-pairs as defined by Berger.

Keywords

p-adic Hodge theorytrianguline representationsB-pairs

MSC classification

Secondary:11F80: Galois representations11F85: $p$-adic theory, local fields11S25: Galois cohomology

Information

Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 4 , July 2009, pp. 865 - 914
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

[1]Bellaïche, J. and Chenevier, G., p-adic families of Galois representations and higher rank Selmer groups, Preprint (2006), arXiv:math.NT/0602340.Google Scholar
[2]Berger, L., Représentations p-adiques et équations différentielles, Invent. Math. 148 (2002), 219–284.CrossRefGoogle Scholar
[3]Berger, L., Equations différentielles p-adiques et (φ,N)-modules filtrés, Astérisque 319 (2008), 13–38.Google Scholar
[4]Berger, L., Construction de (φ,Γ)-modules: représentations p-adiques et B-paires, Algebra Number Theory 2 (2008), 91–120.CrossRefGoogle Scholar
[5]Berger, L. and Breuil, C., Sur quelques représentations potentiellement crystallines de GL2(ℚp), Preprint (2006), arXiv:math/0601545.Google Scholar
[6]Berger, L. and Colmez, P., Familles de représentations de de Rham et monodromie p-adique, Astérisque 319 (2008), 303–337.Google Scholar
[7]Bloch, S. and Kato, K., L-functions and Tamagawa numbers of motives, in The Grothendieck Festschrift, Vol. I, Progress in Mathematics, vol. 86 (Birkhäuser, Boston, 1990), 333–400.Google Scholar
[8]Bushnell, C.-J. and Henniart, G., The local Langlands conjecture for GL(2), Grundlehren der Mathematischen Wissenshaften, vol. 335 (Springer, Berlin, 2006).CrossRefGoogle Scholar
[9]Cherbonnier,   and Colmez, P., Représentations p-adiques surconvergentes, Invent. Math. 133(3) (1998), 581–611.CrossRefGoogle Scholar
[10]Colmez, P., Espaces de Banach de dimension finie, J. Inst. of Math. Jussieu 1 (2002), 331–439.CrossRefGoogle Scholar
[11]Colmez, P., Une correspondance de Langlands locale p-adique pour les représentations semi-stables de dimension 2, Preprint (2004).Google Scholar
[12]Colmez, P., La série principale unitaire de GL2(ℚp), Preprint (2007).Google Scholar
[13]Colmez, P., Représentations triangulines de dimension 2, Astérisque 319 (2008), 187–212.Google Scholar
[14]Colmez, P., Représentations de GL2(ℚp) et (φ,Γ)-modules, Preprint (2008).Google Scholar
[15]Fontaine, J.-M., Représentations p-adiques des corps locaux T, in The Grothendieck Festschrift, Vol. U, Progress in Mathematics, vol. 87 (Birkhäuser, Boston, 1990), 249–309.Google Scholar
[16]Fontaine, J.-M., Le corps des périodes p-adiques. Périodes p-adiques (Bures-sur-Yvette, 1988), Astérisque 223 (1994), 59–111.Google Scholar
[17]Fontaine, J.-M., Représentations p-adiques semi-stables. Périodes p-adiques (Bures-sur-Yvette, 1988), Astérisque 223 (1994), 113–184.Google Scholar
[18]Kedlaya, K., A p-adic local monodromy theorem, Ann. of Math. (2) 160 (2004), 93–184.CrossRefGoogle Scholar
[19]Kedlaya, K., Slope filtrations revisited, Doc. Math. 10 (2005), 447–525 (electronic).CrossRefGoogle Scholar
[20]Kedlaya, K., Some new directions in p-adic Hodge theory, Preprint (2007), arXiv:0709.1970v1.Google Scholar
[21]Kedlaya, K., Slope filtrations for relative Frobenius, Astérisque 319 (2008), 259–301.Google Scholar
[22]Kisin, M., Overconvergent modular forms and the Fontaine–Mazur conjecture, Invent. Math. 153 (2003), 373–454.CrossRefGoogle Scholar
[23]Liu, R., Cohomology and duality for (φ,Γ)-modules over the Robba ring Int. Math. Res. Not. IMRN (3) (2008).CrossRefGoogle Scholar