Publications mathématiques de l'IHÉS

, Volume 130, Issue 1, pp 299–412 | Cite as

A local model for the trianguline variety and applications

  • Christophe BreuilEmail author
  • Eugen Hellmann
  • Benjamin Schraen


We describe the completed local rings of the trianguline variety at certain points of integral weights in terms of completed local rings of algebraic varieties related to Grothendieck’s simultaneous resolution of singularities. We derive several local consequences at these points for the trianguline variety: local irreducibility, description of all local companion points in the crystalline case, combinatorial description of the completed local rings of the fiber over the weight map, etc. Combined with the patched Hecke eigenvariety (under the usual Taylor-Wiles assumptions), these results in turn have several global consequences: classicality of crystalline strictly dominant points on global Hecke eigenvarieties, existence of all expected companion constituents in the completed cohomology, existence of singularities on global Hecke eigenvarieties.


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  1. 1.
    A. Beilinson and J. Bernstein, Localisation de \(\mathfrak{g}\)-modules, C. R. Acad. Sci. Paris, 292 (1981), 15–18. MathSciNetzbMATHGoogle Scholar
  2. 2.
    J. Bellaïche and G. Chenevier, Families of Galois representations and Selmer groups, Astérisque, 324, p. xii+314 (2009). MathSciNetzbMATHGoogle Scholar
  3. 3.
    J. Bellaïche, Critical \(p\)-adic L-functions, Invent. Math., 189 (2012), 1–60. MathSciNetzbMATHGoogle Scholar
  4. 4.
    J. Bergdall, Ordinary modular forms and companion points on the eigencurve, J. Number Theory, 134 (2014), 226–239. MathSciNetzbMATHGoogle Scholar
  5. 5.
    J. Bergdall, Paraboline variation over \(p\)-adic families of \((\varphi ,\Gamma )\)-modules, Compos. Math., 153 (2017), 132–174. MathSciNetzbMATHGoogle Scholar
  6. 6.
    J. Bergdall, Smoothness on definite unitary eigenvarieties at critical points, J. Reine Angew. Math. (2018). CrossRefGoogle Scholar
  7. 7.
    L. Berger, Représentations \(p\)-adiques et équations différentielles, Invent. Math., 148 (2002), 219–284. MathSciNetzbMATHGoogle Scholar
  8. 8.
    L. Berger, Équation différentielles \(p\)-adiques et \((\varphi ,N)\)-modules filtrés, Astérisque, 319 (2008), 13–38. zbMATHGoogle Scholar
  9. 9.
    L. Berger, Constructions de \((\varphi,\Gamma)\)-modules: représentations \(p\)-adiques et \({\rm B}\)-paires, Algebra Number Theory, 2 (2008), 91–120. MathSciNetGoogle Scholar
  10. 10.
    I. N. Bernstein, I. M. Gelfand and S. I. Gelfand, Differential operators on the base affine space and a study of \(\mathfrak{g}\)-modules, Lie groups and their representations, 243 (1975). Google Scholar
  11. 11.
    I. N. Bernstein and A. N. Zelevinsky, Induced representations of reductive \(\mathfrak{p}\)-adic groups. I, Ann. Sci. Éc. Norm. Supér., 10 (1977), 441–472. MathSciNetzbMATHGoogle Scholar
  12. 12.
    R. Bezrukavnikov and S. Riche, Affine braid group actions on derived categories of Springer resolutions, Ann. Sci. Éc. Norm. Supér., 45 (2012), 535–599. MathSciNetzbMATHGoogle Scholar
  13. 13.
    S. Billey and V. Lakshmibai, Singular Loci of Schubert Varieties, Progress in Mathematics, vol. 182, 2000. zbMATHGoogle Scholar
  14. 14.
    S. Bosch and W. Lütkebohmert, Formal and rigid geometry II. Flattening techniques, Math. Ann., 296 (1993), 403–429. MathSciNetzbMATHGoogle Scholar
  15. 15.
    C. Breuil, Vers le socle localement analytique pour \(\mathrm{GL}_{n}\) I, Ann. Inst. Fourier, 66 (2016), 633–685. MathSciNetzbMATHGoogle Scholar
  16. 16.
    C. Breuil, Vers le socle localement analytique pour \(\mathrm{GL}_{n}\) II, Math. Ann., 361 (2015), 741–785. MathSciNetzbMATHGoogle Scholar
  17. 17.
    C. Breuil and M. Emerton, Représentations ordinaires de \({\mathrm {GL}}_{2}(\mathbf {Q}_{p})\) et compatibilité local-global, Astérisque, 331 (2010), 255–315. zbMATHGoogle Scholar
  18. 18.
    C. Breuil and A. Mézard, Multiplicités modulaires et représentations de \({\mathrm {GL}}_{2}(\mathcal{Z}_{p})\) et de \({\mathrm{Gal}}(\overline{\mathbf {Q}_{p}}/\mathbf {Q}_{p})\) en \(\ell=p\), Duke Math. J., 115 (2002), 205–298. MathSciNetGoogle Scholar
  19. 19.
    C. Breuil, E. Hellmann and B. Schraen, Une interprétation modulaire de la variété trianguline, Math. Ann., 367 (2017), 1587–1645. MathSciNetzbMATHGoogle Scholar
  20. 20.
    C. Breuil, E. Hellmann and B. Schraen, Smoothness and classicality on eigenvarieties, Invent. Math., 209 (2017), 197–274. MathSciNetzbMATHGoogle Scholar
  21. 21.
    J. L. Brylinski and M. Kashiwara, Kazhdan-Lusztig conjecture and holonomic systems, Invent. Math., 64 (1981), 387–410. MathSciNetzbMATHGoogle Scholar
  22. 22.
    A. Caraiani, Monodromy and local-global compatibility for \(\ell= p\), Algebra Number Theory, 8 (2014), 1597–1646. MathSciNetzbMATHGoogle Scholar
  23. 23.
    A. Caraiani, M. Emerton, T. Gee, D. Geraghty, V. Paškūnas and S. W. Shin, Patching and the \(p\)-adic local Langlands correspondence, Camb. J. Math., 4 (2016), 197–287. MathSciNetzbMATHGoogle Scholar
  24. 24.
    G. Chenevier, On the infinite fern of Galois representations of unitary type, Ann. Sci. Éc. Norm. Supér., 44 (2011), 963–1019. MathSciNetzbMATHGoogle Scholar
  25. 25.
    G. Chenevier, Sur la densité des representations cristallines de \(\mathrm{Gal}(\overline{\mathbf {Q}_{p}}/\mathbf {Q}_{p})\), Math. Ann., 355 (2015), 1469–1525. MathSciNetGoogle Scholar
  26. 26.
    G. Chenevier, The \(p\)-adic analytic space of pseudocharacters of a profinite group and pseudorepresentations over arbitrary rings, Lond. Math. Soc. Lect. Note Ser., 414 (2014), 221–285. MathSciNetzbMATHGoogle Scholar
  27. 27.
    N. Chriss and V. Ginzburg, Representation Theory and Complex Geometry, Modern, Birkhäuser Classics, 1997. zbMATHGoogle Scholar
  28. 28.
    Y. Ding, Formes modulaires p-adiques sur les courbes de Shimura unitaires et compatibilité local-global, Mémoires Soc. Math. France, vol. 155, 2017. zbMATHGoogle Scholar
  29. 29.
    Y. Ding, Some results on the locally analytic socle for \(\mathrm{GL}_{n}(\mathbf {Q}_{p})\), Int. Math. Res. Not., 287 (2018). Google Scholar
  30. 30.
    Y. Ding, Companion points and locally analytic socle for \({\rm GL}_{2}(L)\), Isr. J. Math., 231 (2019), 47–122. MathSciNetzbMATHGoogle Scholar
  31. 31.
    D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, Graduate Texts in Mathematics, vol. 150, 1995. zbMATHGoogle Scholar
  32. 32.
    M. Emerton, Jacquet modules of locally analytic representations of \(p\)-adic reductive groups I. Construction and first properties, Ann. Sci. Éc. Norm. Supér., 39 (2006), 775–839. MathSciNetzbMATHGoogle Scholar
  33. 33.
    M. Emerton, Jacquet modules of locally analytic representations of \(p\)-adic reductive groups II. The relation to parabolic induction, J. Inst. Math. Jussieu, to appear Google Scholar
  34. 34.
    M. Emerton, Locally Analytic Vectors in Representations of Locally \(p\)-Adic Analytic Groups, Memoirs Amer. Math. Soc., vol. 248, 2017. zbMATHGoogle Scholar
  35. 35.
    M. Emerton and T. Gee, A geometric perspective on the Breuil-Mézard conjecture, J. Inst. Math. Jussieu, 13 (2014), 183–223. MathSciNetzbMATHGoogle Scholar
  36. 36.
    J.-M. Fontaine, Représentations \(\ell\)-adiques potentiellement semi-stables, Astérisque, 223 (1994), 321–347. zbMATHGoogle Scholar
  37. 37.
    J.-M. Fontaine, Arithmétique des représentations galoisiennes \(p\)-adiques, Astérisque, 295 (2004), 1–115. zbMATHGoogle Scholar
  38. 38.
    T. Gee and M. Kisin, The Breuil-Mézard conjecture for potentially Barsotti-Tate representations, Forum Math., Pi, 2, e1 (2014). zbMATHGoogle Scholar
  39. 39.
    V. Ginsburg, \(\mathfrak{G}\)-modules, Springer’s representations and bivariant Chern classes, Adv. Math., 61 (1986), 1–48. MathSciNetzbMATHGoogle Scholar
  40. 40.
    A. Grothendieck and J. Dieudonné, Éléments de géométrie algébrique IV Étude locale des schémas et des morphismes de schémas (première partie), Publ. Math. IHÉS, 20 (1964), 5–259. Google Scholar
  41. 41.
    A. Grothendieck and J. Dieudonné, Éléments de géométrie algébrique IV Étude locale des schémas et des morphismes de schémas (seconde partie), Publ. Math. IHÉS, 24 (1965), 5–231. zbMATHGoogle Scholar
  42. 42.
    A. Grothendieck and J. Dieudonné, Éléments de géométrie algébrique IV Étude locale des schémas et des morphismes de schémas (troisième partie), Publ. Math. IHÉS, 28 (1966), 5–255. Google Scholar
  43. 43.
    R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, vol. 52, 1977. zbMATHGoogle Scholar
  44. 44.
    R. Hotta, K. Takeuchi and T. Tanisaki, \(D\)-Modules, Perverse Sheaves, and Representation Theory, Progress in Mathematics, vol. 236, 2008. zbMATHGoogle Scholar
  45. 45.
    J. Humphreys, Representations of Semisimple Lie Algebras in the BGG Category \(\mathcal{O}\), Graduate Studies in Mathematics, vol. 94, 2008. zbMATHGoogle Scholar
  46. 46.
    J. C. Jantzen, Representations of Algebraic Groups, 2nd ed., Mathematical Surveys and Monographs, vol. 107, 2007. zbMATHGoogle Scholar
  47. 47.
    M. Kashiwara and Y. Saito, Geometric construction of crystal bases, Duke Math. J., 89 (1997), 9–36. MathSciNetzbMATHGoogle Scholar
  48. 48.
    K. Kedlaya, Slope filtrations for relative Frobenius, Astérisque, 319 (2008), 259–301. MathSciNetzbMATHGoogle Scholar
  49. 49.
    K. Kedlaya, J. Pottharst and L. Xiao, Cohomology of arithmetic families of \((\varphi,\Gamma)\)-modules, J. Am. Math. Soc., 27 (2014), 1043–1115. MathSciNetzbMATHGoogle Scholar
  50. 50.
    R. Kiehl and R. Weissauer, Weil Conjectures, Perverse Sheaves and \(\ell\)-adic Fourier Transform, 3rd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete, a Series of Modern Surveys in Mathematics, vol. 42, 2001. zbMATHGoogle Scholar
  51. 51.
    M. Kisin, Overconvergent modular forms and the Fontaine-Mazur conjecture, Invent. Math., 153 (2003), 373–454. MathSciNetzbMATHGoogle Scholar
  52. 52.
    M. Kisin, Potentially semi-stable deformation rings, J. Am. Math. Soc., 21 (2008), 513–546. MathSciNetzbMATHGoogle Scholar
  53. 53.
    M. Kisin, Moduli of finite flat group schemes, and modularity, Ann. Math., 170 (2009), 1085–1180. MathSciNetzbMATHGoogle Scholar
  54. 54.
    R. Liu, Cohomology and duality for \((\varphi,\Gamma)\)-modules over the Robba ring, Int. Math. Res. Not., 32 (2008). Google Scholar
  55. 55.
    R. Liu, Triangulation of refined families, Comment. Math. Helv., 90 (2015), 831–904. MathSciNetzbMATHGoogle Scholar
  56. 56.
    K. Nakamura, Classification of two-dimensional split trianguline representations of \(p\)-adic fields, Compos. Math., 145 (2009), 865–914. MathSciNetzbMATHGoogle Scholar
  57. 57.
    K. Nakamura, Deformations of trianguline \({\rm B}\)-pairs and Zariski density of two dimensional crystalline representations, J. Math. Sci. Univ. Tokyo, 20 (2013), 461–568. MathSciNetzbMATHGoogle Scholar
  58. 58.
    S. Orlik and M. Strauch, On Jordan-Hölder series of some locally analytic representations, J. Am. Math. Soc., 28 (2015), 99–157. zbMATHGoogle Scholar
  59. 59.
    M. Schlessinger, Functors of Artin rings, Trans. Am. Math. Soc., 130 (1968), 208–222. MathSciNetzbMATHGoogle Scholar
  60. 60.
    T. Schmidt and M. Strauch, Dimensions of some locally analytic representations, Represent. Theory, 20 (2016), 14–38. MathSciNetzbMATHGoogle Scholar
  61. 61.
    P. Slodowy, Simple Singularities and Simple Algebraic Groups, Lecture Notes in Math., vol. 815, 1980. zbMATHGoogle Scholar
  62. 62.
    R. Steinberg, On the desingularization of the unipotent variety, Invent. Math., 36 (1976), 209–224. MathSciNetzbMATHGoogle Scholar
  63. 63.
    T. Tanisaki, Characteristic varieties of highest weight modules and primitive quotients, Adv. Stud. Pure Math., 14 (1988), 1–30. MathSciNetzbMATHGoogle Scholar
  64. 64.
    J. Tate, \(p\)-divisible Groups, in Proceedings of a Conference on Local Fields, pp. 158–183, 1966. Google Scholar
  65. 65.
    J. Thorne, On the automorphy of \(\ell\)-adic Galois representations with small residual image, J. Inst. Math. Jussieu, 11 (2012), 855–906. MathSciNetzbMATHGoogle Scholar
  66. 66.
    J. Thorne, A 2-adic automorphy lifting theorem for unitary groups over CM fields, Math. Z., 285 (2017), 1–38. MathSciNetzbMATHGoogle Scholar

Copyright information

© IHES and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Christophe Breuil
    • 1
    Email author
  • Eugen Hellmann
    • 2
  • Benjamin Schraen
    • 3
  1. 1.C.N.R.S., Laboratoire de Mathématique d’Orsay, Université Paris-SudUniversité Paris-SaclayOrsayFrance
  2. 2.Mathematisches InstitutUniversität MünsterMünsterGermany
  3. 3.Laboratoire de Mathématique d’Orsay, Université Paris-SudUniversité Paris-SaclayOrsayFrance

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