Mathematische Annalen

, Volume 367, Issue 3–4, pp 1587–1645 | Cite as

Une interprétation modulaire de la variété trianguline

  • Christophe Breuil
  • Eugen Hellmann
  • Benjamin Schraen
Article

Abstract

Using a patching module constructed in recent work of Caraiani, Emerton, Gee, Geraghty, Paškūnas and Shin we construct some kind of analogue of an eigenvariety. We can show that this patched eigenvariety agrees with a union of irreducible components of a space of trianguline Galois representations. Building on this we discuss the relation with the modularity conjectures for the crystalline case, a conjecture of Breuil on the locally analytic socle of representations occurring in completed cohomology and with a conjecture of Bellaïche and Chenevier on the complete local ring at certain points of eigenvarieties.

Resume

En utilisant le système de Taylor–Wiles–Kisin construit dans un travail récent de Caraiani, Emerton, Gee, Geraghty, Paškūnas et Shin, nous construisons un analogue de la variété de Hecke. Nous montrons que cette variété coïncide avec une union de composantes irréductibles de l’espace des représentations galoisiennes triangulines. Nous précisons les relations de cette construction avec les conjectures de modularité dans le cas cristallin ainsi qu’avec une conjecture de Breuil sur le socle des vecteurs localement analytiques de la cohomologie complétée. Nous donnons également une preuve d’une conjecture de Bellaïche et Chenevier sur l’anneau local complété en certains points des variétés de Hecke.

References

  1. 1.
    Abbes, A.: Éléments de géométrie rigide. Progress in Mathematics, vol. 286. Springer Basel AG, Switzerland (2010)Google Scholar
  2. 2.
    Allen, P.: Deformations of polarized automorphic Galois representations and adjoint Selmer groups. Duke Math. J. arXiv:1411.7661 (à paraître)
  3. 3.
    Barnet-Lamb, T., Gee, T., Geraghty, D., Taylor, R.: Potential automorphy and change of weight. Ann. Math. (2) 179(2), 501–609 (2014)Google Scholar
  4. 4.
    Bellaïche, J., Chenevier, G.: Families of Galois representations and Selmer groups. Astérisque 324 (2009)Google Scholar
  5. 5.
    Bergdall, J.: Paraboline variation over \(p\)-adic families of \((\varphi ,\Gamma )\)-modules. arXiv:1410.3412v1
  6. 6.
    Berger, L.: Représentations \(p\)-adiques et équations différentielles. Invent. Math. 148(2), 219–284 (2002)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Berger, L., Colmez, P.: Familles de représentations de de Rham et monodromie \(p\)-adique. Astérisque 319, 303–337 (2008)Google Scholar
  8. 8.
    Berthelot, P.: Cohomologie rigide et cohomologie rigide à supports propres (prépublication)Google Scholar
  9. 9.
    Bosch, S., Güntzer, U., Remmert, R.: Non-Archimedean analysis. A systematic approach to rigid analytic geometry. In: Grundlehren der Mathematischen Wissenschaften, vol. 261. Springer, Berlin (1984)Google Scholar
  10. 10.
    Breuil, C.: Vers le socle localement analytique pour \({\rm GL}_n\). Annales de l’Institut Fourier (à paraître)Google Scholar
  11. 11.
    Breuil, C.: Vers le socle localement analytique pour \({\rm {GL}}_n\) II. Math. Ann. 361, 741–785 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Breuil, C., Hellmann, E., Schraen, B.: Smoothness and classicality on eigenvarieties. arXiv:1510.01222
  13. 13.
    Breuil, C., Herzig, F.: Ordinary representations of \(G({\mathbb{Q}}_p)\) and fundamental algebraic representations. Duke Math. J. 164, 1271–1352 (2015)Google Scholar
  14. 14.
    Buzzard, K.: Eigenvarieties. In: \(L\)-functions and Galois representations. Based on the symposium, Durham, UK, July 19–30, 2004, pp. 59–120. Cambridge University Press, Cambridge (2007)Google Scholar
  15. 15.
    Caraiani, A.: Local-global compatibility and the action of monodromy on nearby cycles. Duke Math. J. 161(12), 2311–2413 (2012)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Caraiani, A.: Monodromy and local-global compatibility for l = p. Algebra Number Theory 8(7), 1597–1646 (2014)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Caraiani, A., Emerton, M., Gee, T., Geraghty, D., Paškūnas, V., Shin, S.W.: Patching and the \(p\)-adic local langlands correspondence. arXiv:1310.0831
  18. 18.
    Chenevier, G.: Familles \(p\)-adiques de formes automorphes pour \(\text{ GL }_n\). J. Reine Angew. Math. 570, 143–217 (2004)MathSciNetGoogle Scholar
  19. 19.
    Chenevier, G.: Une correspondance de Jacquet–Langlands \(p\)-adique. Duke Math. J. 126(1), 161–194 (2005)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Chenevier, G.: On the infinite fern of Galois representations of unitary type. Ann. Sci. Éc. Norm. Supér. (4) 44(6), 963–1019 (2011)Google Scholar
  21. 21.
    Chenevier, G.: Sur la densité des représentations cristallines de \(\text{ Gal }(\bar{\mathbb{Q}}_p/{\mathbb{Q}}_p)\). Math. Ann. 355(4), 1469–1525 (2013)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Chenevier, G., Harris, M.: Construction of automorphic Galois representations. II. Camb. J. Math. 1(1), 53–73 (2013)Google Scholar
  23. 23.
    Clozel, L., Harris, M., Taylor, R.: Automorphy for some \(l\)-adic lifts of automorphic mod \(l\) Galois representations. Publ. Math. I.H.É.S. 108, 1–181 (2008)Google Scholar
  24. 24.
    Colmez, P.: Représentations triangulines de dimension \(2\). Astérisque 319, 213–258 (2008)Google Scholar
  25. 25.
    Colmez, P., Dospinescu, G.: Complétés universels de représentations de \({\rm {GL}} _2(\mathbb{Q}_p)\). Algebra Number Theory 8, 1447–1519 (2014)Google Scholar
  26. 26.
    Conrad, B.: Irreducible components of rigid spaces. Ann. Inst. Fourier 49(2), 473–541 (1999)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Emerton, M.: Locally analytic vectors in representations of locally \(p\)-adic analytic groups. Mem. AMS (à paraître)Google Scholar
  28. 28.
    Emerton, M.: Jacquet modules of locally analytic representations of \(p\)-adic reductive groups. I. Construction and first properties. Ann. Sci. École Norm. Sup. (4) 39(5), 775–839 (2006)Google Scholar
  29. 29.
    Emerton, M.: On the interpolation of systems of eigenvalues attached to automorphic Hecke eigenforms. Invent. Math. 164(1), 1–84 (2006)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Emerton, M.: Jacquet modules of locally analytic representations of \(p\)-adic reductive groups II. The relation to parabolic induction. J. Inst. Math. de Jussieu (à paraître)Google Scholar
  31. 31.
    Emerton, M., Gee, T.: A geometric perspective on the Breuil–Mézard conjecture. J. Inst. Math. de Jussieu 13, 183–223 (2014)Google Scholar
  32. 32.
    Fontaine, J.-M.: Représentations \(\ell \)-adiques potentiellement semi-stables. Astérisque 223, 321–347 (1994)Google Scholar
  33. 33.
    Grothendieck, A.: Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. I. Pub. Math. I.H.É.S. 20, 5–259 (1964)Google Scholar
  34. 34.
    Hansen, D.: Universal eigenvarieties, trianguline Galois representations, and \(p\)-adic Langlands functoriality. J. Reine Angew. Math. (à paraître)Google Scholar
  35. 35.
    Hellmann, E.: Families of trianguline representations and finite slope spaces. arXiv:1202.4408
  36. 36.
    Hellmann, E.: Families of \(p\)-adic Galois representations and \((\varphi ,\Gamma )\)-modules. arXiv:1202.3413
  37. 37.
    Hellmann, E., Schraen, B.: Density of potentially crystalline representations of fixed weight. Compositio Math. arXiv:1311.3249 (à paraître)
  38. 38.
    Huber, R.: Étale cohomology of rigid analytic varieties and adic spaces. Asp. Math. E30 (1996)Google Scholar
  39. 39.
    Humphreys, J.E.: Representations of Semisimple Lie algebras in the BGG Category \(\cal O\). Graduate Studies in Mathematics, vol. 94. American Mathematical Society, Providence (2008)Google Scholar
  40. 40.
    de Jong, A.J.: Crystalline Dieudonné module theory via formal and rigid geometry. Publ. Math. I.H.É.S. 82, 5–96 (1995)Google Scholar
  41. 41.
    Kedlaya, K., Pottharst, J., Xiao, L.: Cohomology of arithmetic families of \((\phi,\Gamma )\)-modules. J. Am. Math. Soc. 27, 1043–1115 (2014)MathSciNetCrossRefMATHGoogle Scholar
  42. 42.
    Kiehl, R.: Theorem A und Theorem B in der nichtarchimedischen Funktionentheorie. Invent. Math. 2, 256–273 (1967)MathSciNetCrossRefMATHGoogle Scholar
  43. 43.
    Kisin, M.: Overconvergent modular forms and the Fontaine–Mazur conjecture. Invent. Math. 153(2), 373–454 (2003)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    Kisin, M.: Potentially semi-stable deformation rings. J. Am. Math. Soc. 21(2), 513–546 (2008)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    Kisin, M.: Moduli of finite flat group schemes, and modularity. Ann. Math. (2) 170(3), 1085–1180 (2009)Google Scholar
  46. 46.
    Kudla, S.S.: The local Langlands correspondence: the non-Archimedean case. In: Motives (Seattle, WA, 1991), Proc. Sympos. Pure Math., vol. 55, pp. 365–391. American Mathematical Society, Providence (1994)Google Scholar
  47. 47.
    Labesse, J.-P.: Changement de base CM et séries discrètes. In: On the Stabilization of the Trace Formula, pp. 429–470. International Press, Somerville (2011)Google Scholar
  48. 48.
    Lazard, M.: Groupes analytiques \(p\)-adiques. Pub. Math. I.H.É.S. 26, 5–219 (1965)Google Scholar
  49. 49.
    Liu, R.: Triangulation of refined families. Comment. Math. Helv. (à paraître)Google Scholar
  50. 50.
    Mœglin, C., Waldspurger, J.-L.: Le spectre résiduel de \({\rm {GL}}(n)\). Annales scientifiques de l’É.N.S.22, 605–674 (1989)Google Scholar
  51. 51.
    Orlik, S., Strauch, M.: On Jordan–Hölder series of some locally analytic representations. J. Am. Math. Soc. 28, 99–157 (2015)CrossRefMATHGoogle Scholar
  52. 52.
    Schneider, P.: Nonarchimedean Functional Analysis. Springer Monographs in Mathematics. Springer, Berlin (2002)Google Scholar
  53. 53.
    Schneider, P., Teitelbaum, J.: Banach space representations and Iwasawa theory. Isr. J. Math. 127, 359–380 (2002)MathSciNetCrossRefMATHGoogle Scholar
  54. 54.
    Schneider, P., Teitelbaum, J.: Locally analytic distributions and \(p\)-adic representation theory, with applications to \({\rm {GL}}_2\). J. Am. Math. Soc. 15, 443–468 (2001)MathSciNetCrossRefMATHGoogle Scholar
  55. 55.
    Schneider, P., Teitelbaum, J.: \(p\)-adic Fourier theory. Doc. Math. 6, 447–481 (2001)MathSciNetMATHGoogle Scholar
  56. 56.
    Schneider, P., Teitelbaum, J.: Algebras of \(p\)-adic distributions and admissible representations. Invent. Math. 153, 145–196 (2003)MathSciNetCrossRefMATHGoogle Scholar
  57. 57.
    Schneider, P., Teitelbaum, J.: Banach–Hecke algebras and \(p\)-adic Galois representations. Doc. Math. Extra Vol., 631–684 (2006)Google Scholar
  58. 58.
    Schneider, P., Teitelbaum, J.: Duality for admissible locally analytic representations. Represent. Theory 9, 297–326 (2005)MathSciNetCrossRefMATHGoogle Scholar
  59. 59.
    Schraen, B.: Représentations localement analytiques de \({\rm {GL}}_3({\mathbb{Q}}_p)\). Ann. Sci. Éc. Norm. Supér 44, 43–145 (2011)Google Scholar
  60. 60.
    Serre, J.-P.: Cohomologie Galoisienne. Lecture Notes in Mathematics, vol. 5, Springer, Berlin (1994)Google Scholar
  61. 61.
    Thorne, J.: On the automorphy of \(\ell \)-adic Galois representations with small residual image. J. Inst. Math. Jussieu 11, 855–920 (2012)Google Scholar
  62. 62.
    Zelevinsky, A.V.: Induced representations of reductive \({\mathfrak{p}}\)-adic groups. II. On irreducible representations of \({\rm {GL}} (n)\). Ann. Sci. École Norm. Sup. (4) 13(2), 165–210 (1980)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Christophe Breuil
    • 1
  • Eugen Hellmann
    • 2
  • Benjamin Schraen
    • 3
  1. 1.Faculté d’OrsayC.N.R.S. Université Paris-SudOrsay CedexFrance
  2. 2.Mathematisches InstitutUniversität BonnBonnGermany
  3. 3.CMLS, École polytechnique, CNRSUniversité Paris-SaclayPalaiseau CédexFrance

Personalised recommendations