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The image of Colmez’s Montreal functor

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Abstract

We prove a conjecture of Colmez concerning the reduction modulo p of invariant lattices in irreducible admissible unitary p-adic Banach space representations of GL2(Q    p ) with p≥5. This enables us to restate nicely the p-adic local Langlands correspondence for GL2(Q    p ) and deduce a conjecture of Breuil on irreducible admissible unitary completions of locally algebraic representations.

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References

  1. L. Barthel and R. Livné, Irreducible modular representations of GL2 of a local field, Duke Math. J., 75 (1994), 261–292.

    Article  MathSciNet  MATH  Google Scholar 

  2. J. Bellaïche, Pseudodeformations, Math. Z., 270 (2012), 1163–1180.

    Article  MathSciNet  MATH  Google Scholar 

  3. J. Bellaïche and G. Chenevier, Families of Galois Representations and Selmer Groups, Astérisque, vol. 324, Soc. Math. France, Paris, 2009.

    MATH  Google Scholar 

  4. L. Berger, Représentations modulaires de GL2(Q    p ) et représentations galoisiennes de dimension 2, Astérisque, 330 (2010), 263–279.

    Google Scholar 

  5. L. Berger, La correspondance de Langlands locale p-adique pour GL2(Q    p ), in Exposé No 1017 du Séminaire Bourbaki, Astérisque, vol. 339, pp. 157–180, 2011.

    Google Scholar 

  6. L. Berger, Central characters for smooth irreducible modular representations of GL2(Q    p ), Rendiconti del Seminario Matematico della Università di Padova (F. Baldassarri’s 60th birthday), vol. 127, 2012.

  7. L. Berger and C. Breuil, Sur quelques représentations potentiellement cristallines de GL2(Q    p ), Astérisque, 330 (2010), 155–211.

    MathSciNet  Google Scholar 

  8. J.-N. Bernstein (rédigé par P. Deligne), Le ‘centre’ de Bernstein, in Représentations des groupes réductifs sur un corps local, pp. 1–32, Herman, Paris, 1984.

  9. G. Böckle, Demuškin groups with group actions and applications to deformations of Galois representations, Compositio, 121 (2000), 109–154.

    Article  MATH  Google Scholar 

  10. N. Boston, H. W. Lenstra, and K. A. Ribet, Quotients of group rings arising from two dimensional representations, C.R. Acad. Sci. Paris Sér. I, 312 (1991), 323–328.

    MathSciNet  MATH  Google Scholar 

  11. N. Bourbaki, Algébre, Chapitre 8, Hermann, Paris, 1958.

    Google Scholar 

  12. N. Bourbaki, Commutative Algebra, Hermann, Paris, 1972.

    MATH  Google Scholar 

  13. N. Bourbaki, Algébre Homologique, Masson, Paris, 1980.

    MATH  Google Scholar 

  14. N. Bourbaki, Algebra I, Chapters 1–3, Springer, Berlin, 1989.

    MATH  Google Scholar 

  15. C. Breuil and A. Mézard, Multiplicités modulaires et représentations de GL2(Z p ) et de \(\mathrm{Gal}(\overline{\mathbf {Q}}_{\,\,p}/\mathbf {Q}_{\,\,p})\) en l=p, Duke Math. J., 115 (2002), 205–310.

    Article  MathSciNet  MATH  Google Scholar 

  16. C. Breuil, Sur quelques représentations modulaires et p-adiques de GL2(Q    p ). I, Compositio, 138 (2003), 165–188.

    Article  MathSciNet  MATH  Google Scholar 

  17. C. Breuil, Sur quelques représentations modulaires et p-adiques de GL2(Q    p ). II, J. Inst. Math. Jussieu, 2 (2003), 1–36.

    Article  MathSciNet  Google Scholar 

  18. C. Breuil, Invariant \(\mathcal{L}\) et série spéciale p-adique, Ann. Sci. Éc. Norm. Super., 37 (2004), 559–610.

    MathSciNet  MATH  Google Scholar 

  19. C. Breuil and M. Emerton, Représentations p-adiques ordinaires de GL2(Q    p ) et compatibilité local-global, Astérisque, 331 (2010), 255–315.

    MathSciNet  Google Scholar 

  20. C. Breuil and V. Paškūnas, Towards a Modulo p Langlands Correspondence for GL2, Memoirs of AMS, vol. 216, 2012.

    Google Scholar 

  21. A. Brumer, Pseudo-compact algebras, profinite groups and class formations, J. Algebra, 4 (1966), 442–470.

    Article  MathSciNet  MATH  Google Scholar 

  22. G. Chenevier, The p-adic analytic space of pseudocharacters of a profinite group, and pseudorepresentations over arbitrary rings, arXiv:0809.0415.

  23. P. Colmez, Représentations de GL2(Q    p ) et (φ,Γ)-modules, Astérisque, 330 (2010), 281–509.

    MathSciNet  Google Scholar 

  24. C. W. Curtis and I. Reiner, Methods of Representation Theory. Volume I, Wiley, New York, 1981.

    Google Scholar 

  25. M. Demazure and P. Gabriel, Groupes Algébriques, Tome I, Masson, Paris, 1970.

    MATH  Google Scholar 

  26. M. Demazure and A. Grothendieck, Schémas en Groupes I, Lect. Notes Math, vol. 151, Springer, Berlin, 1970.

    MATH  Google Scholar 

  27. G. Dospinescu and B. Schraen, Endomorphism algebras of p-adic representations of p-adic Lie groups, arXiv:1106.2442.

  28. M. Emerton, p-adic L-functions and unitary completions of representations of p-adic reductive groups, Duke Math. J., 130 (2005), 353–392.

    MathSciNet  MATH  Google Scholar 

  29. M. Emerton, A local-global compatibility conjecture in the p-adic Langlands programme for GL2/Q, Pure Appl. Math. Q., 2 (2006), 279–393.

    Article  MathSciNet  MATH  Google Scholar 

  30. M. Emerton, Ordinary parts of admissible representations of p-adic reductive groups I. Definition and first properties, Astérisque, 331 (2010), 335–381.

    Google Scholar 

  31. M. Emerton, Ordinary parts of admissible representations of p-adic reductive groups II. Derived functors, Astérisque, 331 (2010), 383–438.

    Google Scholar 

  32. M. Emerton, Local-global compatibility in the p-adic Langlands programme for GL2/Q.

  33. M. Emerton, Locally analytic vectors in representations of locally p-adic analytic groups, Memoirs of the AMS, to appear.

  34. M. Emerton and V. Paškūnas, On effaceability of certain δ-functors, Astérisque, 331 (2010), 439–447.

    Google Scholar 

  35. P. Gabriel, Des catégories abéliennes, Bull. Soc. Math. Fr., 90 (1962), 323–448.

    MathSciNet  MATH  Google Scholar 

  36. E. Ghate and A. Mézard, Filtered modules with coefficients, Trans. Am. Math. Soc., 361 (2009), 2243–2261.

    Article  MATH  Google Scholar 

  37. Y. Hu, Sur quelques représentations supersinguliéres de \(\mathrm{GL}_{2}(\mathbf {Q}_{\,p^{f}})\), J. Algebra, 324 (2010), 1577–1615.

    Article  MathSciNet  MATH  Google Scholar 

  38. I. Kaplansky, Commutative Rings, revised ed., University of Chicago Press, Chicago, 1974.

    MATH  Google Scholar 

  39. M. Kisin, Moduli of finite flat group schemes and modularity, Ann. Math., 170 (2009), 1085–1180.

    Article  MathSciNet  MATH  Google Scholar 

  40. M. Kisin, The Fontaine-Mazur conjecture for GL2, J. Am. Math. Soc., 22 (2009), 641–690.

    Article  MathSciNet  MATH  Google Scholar 

  41. M. Kisin, Deformations of \(G_{\mathbf {Q}_{\,\,p}}\) and GL2(Q    p ) representations, Astérisque, 330 (2010), 511–528.

    MathSciNet  Google Scholar 

  42. J. Labute, Classification of Demuškin groups, Can. J. Math., 19 (1967), 106–132.

    Article  MathSciNet  MATH  Google Scholar 

  43. T. Y. Lam, A First Course in Noncommutative Rings, Springer GTM, vol. 131, 1991.

    Book  MATH  Google Scholar 

  44. S. Lang, Algebra, revised 3rd ed., Springer, Berlin, 2002.

    Book  MATH  Google Scholar 

  45. M. Lazard, Groupes analytiques p-adiques, Publ. Math. IHES 26 (1965).

  46. H. Matsumura, Commutative ring theory, CUP.

  47. B. Mazur, Deforming Galois representations, in Y. Ihara, K. Ribet, J.-P. Serre (eds.), Galois Groups over Q, Berkley, CA, 1987, pp. 385–437. Springer, New York, 1989.

    Chapter  Google Scholar 

  48. B. Mazur, An introduction to the deformation theory of Galois representations, in G. Cornell, J. H. Silverman, and G. Stevens (eds.) Modular Forms and Fermat’s Last Theorem, Boston, MA, 1995, pp. 243–311. Springer, New York, 1997.

    Chapter  Google Scholar 

  49. J. Neukirch, A. Schmidt, and K. Wingberg, Cohomology of Number Fields, Springer, Berlin, 2000.

    MATH  Google Scholar 

  50. L. Nyssen, Pseudo-représentations, Math. Ann., 306 (1996), 257–283.

    Article  MathSciNet  MATH  Google Scholar 

  51. R. Ollivier, Le foncteur des invariants sous l’action du pro-p-Iwahori de GL(2,F), J. Reine Angew. Math., 635 (2009), 149–185.

    MathSciNet  MATH  Google Scholar 

  52. F. Oort, Yoneda extensions in abelian categories, Math. Ann., 153 (1964), 227–235.

    Article  MathSciNet  MATH  Google Scholar 

  53. D. Prasad, Locally algebraic representations of p-adic groups, appendix to [60].

  54. V. Paškūnas, Coefficient Systems and Supersingular Representations of GL2(F), Mémoires de la SMF, vol. 99, 2004.

    Google Scholar 

  55. V. Paškūnas, On some crystalline representations of GL2(Q    p ), Algebra Number Theory, 3 (2009), 411–421.

    Article  MathSciNet  MATH  Google Scholar 

  56. V. Paškūnas, Extensions for supersingular representations of GL2(Q    p ), Astérisque, 331 (2010), 317–353.

    Google Scholar 

  57. V. Paškūnas, Admissible unitary completions of locally Q    p -rational representations of GL2(F), Represent. Theory, 14 (2010), 324–354.

    Article  MathSciNet  MATH  Google Scholar 

  58. V. Paškūnas, Blocks for mod p representations of GL2(Q    p ), preprint (2011), arXiv:1104.5602.

  59. P. Schneider, Nonarchimedean Functional Analysis, Springer, Berlin, 2001.

    Google Scholar 

  60. P. Schneider and J. Teitelbaum, \(U(\mathfrak{g})\)-finite locally analytic representations, Represent. Theory, 5 (2001), 111–128.

    Article  MathSciNet  MATH  Google Scholar 

  61. P. Schneider and J. Teitelbaum, Banach space representations and Iwasawa theory, Isr. J. Math., 127 (2002), 359–380.

    Article  MathSciNet  MATH  Google Scholar 

  62. J.-P. Serre, Sur la dimension cohomologique des groupes profinis, Topology, 3 (1965), 413–420.

    Article  MathSciNet  MATH  Google Scholar 

  63. J.-P. Serre, Linear Representation of Finite Groups, Springer, Berlin, 1977.

    Book  Google Scholar 

  64. J.-P. Serre, Cohomologie Galoisienne, Cinquième édition, révisée et complétée, Springer, Berlin, 1997.

    Book  Google Scholar 

  65. R. Taylor, Galois representations associated to Siegel modular forms of low weight, Duke Math. J., 63 (1991), 281–332.

    Article  MathSciNet  MATH  Google Scholar 

  66. M. van den Bergh, Blowing up of non-commutative smooth surfaces, Mem. Amer. Math. Soc., 154 (2001), no. 734.

    Google Scholar 

  67. M.-F. Vignéras, Representations modulo p of the p-adic group GL(2,F), Compos. Math., 140 (2004), 333–358.

    Article  MathSciNet  MATH  Google Scholar 

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  1. Fakultät für Mathematik, Universität Duisburg-Essen, 45127, Essen, Germany

    Vytautas Paškūnas

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Paškūnas, V. The image of Colmez’s Montreal functor. Publ.math.IHES 118, 1–191 (2013). https://doi.org/10.1007/s10240-013-0049-y

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  • DOI: https://doi.org/10.1007/s10240-013-0049-y

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