Advertisement

On crystabelline deformation rings of \(\mathrm {Gal}(\overline{\mathbb {Q}}_p/\mathbb {Q}_p)\) (with an appendix by Jack Shotton)

  • Yongquan Hu
  • Vytautas Paškūnas
Article

Abstract

We prove that certain crystabelline deformation rings of two dimensional residual representations of \(\mathrm {Gal}(\overline{\mathbb {Q}}_p/\mathbb {Q}_p)\) are Cohen–Macaulay. As a consequence, this allows to improve Kisin’s \(R[1/p]=\mathbb {T}[1/p]\) theorem to an \(R=\mathbb {T}\) theorem.

Mathematics Subject Classification

11F80 11F85 

Notes

Acknowledgements

YH was partially supported by National Natural Science Foundation of China Grants 11688101; China’s Recruitement Program of Global Experts, National Center for Mathematics and Interdisciplinary Sciences and Hua Loo-Keng Center for Mathematical Sciences of Chinese Academy of Sciences. VP was partially supported by SFB/TR45 of DFG. The project started when YH visited VP in 2013 supported by SFB/TR45 and he would like to thank the University Duisburg-Essen for the invitation and the hospitality. The authors would like to thank Jack Shotton for the appendix to the paper, as well as Toby Gee, James Newton, Shu Sasaki and Jack Thorne for their comments. We also thank the anonymous referee for their careful reading of the paper and pertinent comments.

References

  1. 1.
    Allen, P.B.: Deformations of polarized automorphic Galois representations and adjoint Selmer groups. Duke Math. J. 165(13), 2407–2460 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Barnet-Lamb, T., Geraghty, D., Harris, M., Taylor, R.: A family of Calabi–Yau varieties and potential automorphy II. Publ. Res. Inst. Math. Sci. 47(1), 29–98 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Barthel, L., Livné, R.: Irreducible modular representations of \({\text{ GL }}_2\) of a local field. Duke Math. J. 75, 261–292 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Berger, L., Breuil, C.: Sur quelques représentations potentiellement cristallines de \({\text{ GL }}_2({\mathbb{Q}}_p)\). Astérisque 330, 155–211 (2010)zbMATHGoogle Scholar
  5. 5.
    Breuil, C.: Sur quelques représentations modulaires et \(p\)-adiques de \({\text{ GL }}_2({\mathbb{Q}}_p)\): I. Compos. Math. 138, 165–188 (2003)CrossRefGoogle Scholar
  6. 6.
    Breuil, C.: Sur quelques représentations modulaires et \(p\)-adiques de \({\text{ GL }}_2({\mathbb{Q}}_p)\): II. J. Inst. Math. Jussieu 2, 1–36 (2003)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Breuil, C., Mézard, A.: Multiplicités modulaires et représentations de \({\text{ GL }}_2({\mathbb{Z}}_p)\) et de \({\text{ Gal }}( {{\overline{\mathbb{Q}}}_p}/{\mathbb{Q}}_p)\) en \(l=p\). Duke Math. J. 115, 205–310 (2002)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Breuil C., Paškūnas V.: Towards a Mod \(p\) Langlands Correspondence for \({\text{ GL }}_2\), vol. 216. Memoirs of American Mathematical Society(2012)Google Scholar
  9. 9.
    Bushnell, C.J., Kutzko, P.C.: Smooth representations of \(p\)-adic reductive groups: structure theory via types. Proc. Lond. Math. Soc. 3(77), 582–634 (1998)CrossRefzbMATHGoogle Scholar
  10. 10.
    Buzzard, K., Diamond, F., Jarvis, F.: On Serre’s conjecture for mod \(\ell \) Galois representations over totally real fields. Duke Math. J. 55, 105–161 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Caraiani, A., Emerton, M., Gee, T., Geraghty, D., Paškūnas, V., Shin, S.W.: Patching and the \(p\)-adic Langlands program for \({\text{ GL }}(2, {\mathbb{Q}}_p)\). Compos. Math. 154(3), 503–548 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Colmez, P.: Représentations de \({\text{ GL }}_2({\mathbb{Q}}_p)\) et \((\varphi,\Gamma )\)-modules. Astérisque 330, 281–509 (2010)Google Scholar
  13. 13.
    Emerton, M., Gee, T.: A geometric perspective on the Breuil-Mézard conjecture. J. Inst. Math. Jussieu 13(1), 183–223 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Emerton, M., Gee, T., Savitt, D.: Lattices in the cohomology of Shimura curves. Invent. Math. 200, 1–96 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Eisenbud, D.: Commutative Algebra with a View Toward Algebraic Geometry, GTM, vol. 150. Springer, Berlin (2008)Google Scholar
  16. 16.
    Fontaine, J.-M.: Représentations \(p\)-adiques semi-stables. Astérisque 223, 113–184 (1994)zbMATHGoogle Scholar
  17. 17.
    Galatius, S., Venkatesh, A.: Derived Galois deformation rings. Adv. Math. 327, 470–623 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Gee T.: Modularity lifting theorems—notes for Arizona Winter School. http://wwwf.imperial.ac.uk/~tsg/
  19. 19.
    Gee, T., Kisin, M.: The Breuil–Mézard conjecture for potentially Barsotti–Tate representations. Forum Math. Pi 2, e1, 56 (2014)CrossRefzbMATHGoogle Scholar
  20. 20.
    Glover, D.J.: A study of certain modular representations. J. Algebra 51, 425–475 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Helm D.: Curtis homomorphisms and the integral Bernstein centre for \({\text{ GL }}_n\), Preprint (2016). arXiv:1605.00487
  22. 22.
    Henniart G.: Sur l’unicité des types pour \({\text{ GL }}_2\), Appendix to [7]Google Scholar
  23. 23.
    Hu, Y., Tan, F.: The Breuil–Mézard conjecture for split non-scalar residual representations. Ann. Sci. Éc. Norm. Supér. (4) 48(6), 1383–1421 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Khare, C., Wintenberger, J.-P.: Serre’s modularity conjecture (I). Invent. Math. 178, 485–504 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Khare, C., Wintenberger, J.-P.: Serre’s modularity conjecture (II). Invent. Math. 178, 505–586 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Khare, C., Wintenberger, J.-P.: On Serre’s conjecture for \(2\)-dimensional mod \(p\) representations of \({\text{ Gal }}(\overline{{\mathbb{Q}}}/{\mathbb{Q}})\). Ann. Math. 169, 229–253 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kisin, M.: Modularity of \(2\)-Dimensional Galois Representations, Current Developments in Mathematics, 2005, 191–230. International Press, Somerville (2007)Google Scholar
  28. 28.
    Kisin, M.: Modularity of \(2\)-adic Barsotti–Tate representations. Invent. Math. 178, 587–634 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Kisin, M.: Potentially semi-stable deformation rings. J. Am. Math. Soc. 21(2), 513–546 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Kisin, M.: The Fontaine–Mazur conjecture for \({\text{ GL }}_2\). J. Am. Math. Soc. 22(3), 641–690 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Matsumura, H.: Commutative Ring Theory. Cambridge Studies in Advanced Mathematics 8, 2nd edn. CUP, Cambridge (1989)zbMATHGoogle Scholar
  32. 32.
    Mazur, B.: Deforming Galois Representations. In: Ihara, Y., Ribet, K., Serre, J.-P. (eds) Galois Groups Over \({\overline{\mathbb{Q}}}_p\). Mathematical Sciences Research Institute Publications. vol. 16, pp. 385–437 (1987)Google Scholar
  33. 33.
    Morra, S.: Explicit description of irreducible \({\text{ GL }}_2({\mathbb{Q}}_p)\)-representations over \(\overline{{\mathbb{F}}}_p\). J. Algebra 339, 252–303 (2011)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Morra, S.: Invariant elements for \(p\)-modular representations of \({\text{ GL }}_2({\mathbb{Q}}_p)\). Trans. Am. Math. Soc. 365, 6625–6667 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Paškūnas, V.: On some crystalline representations of \({\text{ GL }}_2({\mathbb{Q}}_p)\). Algebra Number Theory 3(4), 411–421 (2009)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Paškūnas, V.: Admissible unitary completions of locally \({\mathbb{Q}}_p\)-rational representations of \({\text{ GL }}_2(F)\). Represent. Theory Am. Math. Soc. 14, 324–354 (2010)CrossRefzbMATHGoogle Scholar
  37. 37.
    Paškūnas, V.: The image of Colmez’s Montreal functor. Publ. Math. Inst. Hautes Études Sci. 118, 1–191 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Paškūnas, V.: Blocks for mod \(p\) representations of \({\text{ GL }}_2({mathbb{Q}}_p)\). In: Automorphic Forms And Galois Representations. Vol. 2. London Math. Soc. Lecture Note Ser., vol. 415, pp. 231–247 (2014). CUPGoogle Scholar
  39. 39.
    Paškūnas, V.: On the Breuil–Mézard Conjecture. Duke Math. J. 164(2), 297–359 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Paškūnas, V.: On \(2\)-adic deformations. Math. Z. 286(3–4), 801–819 (2017)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Paškūnas, V.: On \(2\)-dimensional \(2\)-adic Galois representations of local and global fields. Algebra Number Theory 10(6), 1301–1358 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Pilloni V.: The study of \(2\)-dimensional \(p\)-adic Galois deformations in the \(l\ne p\) case, preprint (2008). http://perso.ens-lyon.fr/vincent.pilloni/Defo.pdf
  43. 43.
    Sander, F.: Hilbert–Samuel multiplicities of certain deformation rings. Math. Res. Lett. 21(3), 605–615 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Schneider, P., Stuhler, U.: Representation theory and sheaves on the Bruhat-Tits building. Publ. Math. Inst. Hautes Études Sci. 85, 97–191 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Schraen, B.: Sur la présentation des représentations supersingulières de \({\text{ GL }}_2(F)\). J. Reine Angew. Math. 704, 187–208 (2015)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Shotton, J.: Local deformation rings for \({\text{ GL }}_2\) and a Breuil–Mézard conjecture when \(l \ne p\). Algebra Number Theory 10(7), 1437–1475 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Shotton, J.: The Breuil–Mézard conjecture when \(l\ne p\). Duke Math. J. 167(4), 603–678 (2018)MathSciNetCrossRefGoogle Scholar
  48. 48.
    Snowden A.: Singularities of ordinary deformation rings, preprint (2011). arXiv:1111.3654
  49. 49.
    The Stacks Project Authors: Stacks Project (2017). http://stacks.math.columbia.edu
  50. 50.
    Taylor R.: On the meromorphic continuation of degree two \(L\)-functions. Documenta Mathematica, Extra Volume: John Coates’ Sixtieth Birthday, pp. 729–779 (2006)Google Scholar
  51. 51.
    Taylor, R.: On icosahedral Artin representations. II. Am. J. Math. 125, 549–566 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Vignéras, M.-F.: A criterion for integral structures and coefficient systems on the tree of \({\rm PGL}(2, F)\). Pure Appl. Math. Q. 4(4), 1291–1316 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Yoshino, Y.: Cohen–Macaulay Modules Over Cohen–Macaulay Rings. London Mathematical Society Lecture Note Series, vol. 146. CUP, Cambridge (1990)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Morningside Center of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of SciencesUniversity of the Chinese Academy of SciencesBeijingChina
  2. 2.Fakultät für MathematikUniversität Duisburg-EssenEssenGermany

Personalised recommendations