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Extensions of rank one (φ,Γ)-modules and crystalline representations

Part of: Algebraic number theory: local and $p$-adic fields

Published online by Cambridge University Press:  13 December 2010

Seunghwan Chang  and
Fred Diamond
Seunghwan Chang
Affiliation:
Institute of Mathematical Sciences, Ewha Womans University, Seoul 120-750, Republic of Korea (email: schang@ewha.ac.kr)
Fred Diamond
Affiliation:
Department of Mathematics, King’s College London, Strand, London WC2R 2LS, UK (email: Fred.Diamond@kcl.ac.uk)
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Abstract

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Let K be a finite unramified extension of Qp. We parametrize the (φ,Γ)-modules corresponding to reducible two-dimensional -representations of GK and characterize those which have reducible crystalline lifts with certain Hodge–Tate weights.

Keywords

p-adic representationscrystalline representations(φ,Γ)-modulesWach modules

MSC classification

Secondary: 11S25: Galois cohomology

Information

Type
Research Article
Information
Compositio Mathematica , Volume 147 , Issue 2 , March 2011 , pp. 375 - 427
Copyright
Copyright © Foundation Compositio Mathematica 2010

References

[1]Berger, L., Représentations p-adiques et équations différentielles, Invent. Math. 148 (2002), 219–284.CrossRefGoogle Scholar
[2]Berger, L., An introduction to the theory of p-adic representations, in Geometric aspects of Dwork theory (Walter de Gruyter, Berlin, 2004), 255–292.CrossRefGoogle Scholar
[3]Berger, L., Limites de représentations cristallines, Compositio Math. 140 (2004), 1473–1498.CrossRefGoogle Scholar
[4]Breuil, C., Sur un problème de compatibilité local–global modulo p pour GL 2. Preprint (2009),http://www.math.u-psud.fr/breuil/PUBLICATIONS/compamodp.pdf.Google Scholar
[5]Buzzard, K., Diamond, F. and Jarvis, F., On Serre’s conjecture for mod  ℓ Galois representations over totally real fields, Duke Math. J. 155 (2010), 105–161.CrossRefGoogle Scholar
[6]Chang, S., Extensions of rank one (φ,Γ)-modules, PhD thesis, Brandeis University (2006).Google Scholar
[7]Cherbonnier, F. and Colmez, P., Représentations p-adiques surconvergentes, Invent. Math. 133 (1998), 581–611.CrossRefGoogle Scholar
[8]Colmez, P., Représentations cristallines et représentations de hauteur finie, J. Reine Angew. Math. 514 (1999), 119–143.CrossRefGoogle Scholar
[9]Colmez, P. and Fontaine, J.-M., Constructions des représentations p-adiques semi-stables, Invent. Math. 140 (2000), 1–43.CrossRefGoogle Scholar
[10]Dousmanis, G., On reductions of families of crystalline Galois representations. Preprint (2008), arXiv:0805.1634.Google Scholar
[11]Fontaine, J.-M., Représentations p-adiques des corps locaux I, in The Grothendieck Festschrift, vol. II (Birkhäuser, Boston, 1990), 249–309.Google Scholar
[12]Fontaine, J.-M., Le corps des périodes p-adiques, in Périodes p-adiques (Bures-sur-Yvette, 1988), Astérisque 223 (1994), 59–102.Google Scholar
[13]Fontaine, J.-M., Représentations p-adiques semi-stables, in Périodes p-adiques (Bures-sur-Yvette, 1988), Astérisque 223 (1994), 113–184.Google Scholar
[14]Herr, L., Sur la cohomologie galoisienne des corps p-adiques, Bull. Soc. Math. France 126 (1998), 563–600.CrossRefGoogle Scholar
[15]Herr, L., Une approche nouvelle de la dualité locale de Tate, Math. Ann. 320 (2001), 307–337.CrossRefGoogle Scholar
[16]Khare, C. and Wintenberger, J.-P., On Serre’s conjecture for 2-dimensional mod  p representations of , Ann. of Math. (2) 169 (2009), 229–253.CrossRefGoogle Scholar
[17]Khare, C. and Wintenberger, J.-P., Serre’s modularity conjecture (II), Invent. Math. 178 (2009), 505–586.CrossRefGoogle Scholar
[18]Liu, R., Cohomology and duality for (φ,Γ)-modules over the Robba ring, Int. Math. Res. Not. IMRN 3 (2008), 32pp, Art. ID rnm150.Google Scholar
[19]Wach, N., Représentations p-adiques potentiellement cristallines, Bull. Soc. Math. France 124 (1996), 375–400.CrossRefGoogle Scholar
[20]Wach, N., Représentations cristallines de torsion, Compositio Math. 108 (1997), 185–240.CrossRefGoogle Scholar