Skip to main content
SpringerLink
Log in

On the modularity of 2-adic potentially semi-stable deformation rings

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

Using p-adic local Langlands correspondence for \({\text {GL}}_2(\mathbb {Q}_2)\) and an ordinary \(R = \mathbb {T}\) theorem, we prove that the support of patched modules for quaternionic forms meet every irreducible component of the potentially semi-stable deformation ring. This gives a new proof of the Breuil–Mézard conjecture for 2-dimensional representations of the absolute Galois group of \(\mathbb {Q}_2\), which is new in the case \(\overline{r}\) a twist of an extension of the trivial character by itself. As a consequence, a local restriction in the proof of Fontaine-Mazur conjecture in Paškūnas (Algebra Number Theory 10(6):1301–1358, 2016) is removed.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Allen, P.B.: Deformations of Hilbert modular Galois representations and adjoint selmer groups (2014). https://faculty.math.illinois.edu/~pballen/Smooth.pdf

  2. Allen, P.B.: Modularity of nearly ordinary 2-adic residually dihedral Galois representations. Compos. Math. 150(8), 1235–1346 (2014)

    MathSciNet  MATH  Google Scholar 

  3. Babnik, M.: Irreduzible Komponenten von 2-adischen Deformationsräumen (2015). arXiv:1512.09277

  4. Berger, L., Breuil, C.: Sur quelques représentations potentiellement cristallines de \( \text{ GL}_2(\mathbb{Q}_p)\). Astérisque 330, 155–211 (2010)

    MATH  Google Scholar 

  5. Breuil, C.: Emerton, M.: Représentations \(p\)-adiques ordinaires de \( \text{ GL}_2(\mathbb{Q}_p)\) et compatibilité local-global. Astérisque 331, 255–315 (2010)

  6. Bushnell, C.J., Henniart, G.: The local Langlands conjecture for \(\text{ GL }(2)\). Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 335. Springer, Berlin (2006)

  7. Breuil, C., Herzig, F.: Ordinary representations of \(G(\mathbb{Q}_p)\) and fundamental algebraic representations. Duke Math. J. 164(7), 1271–1352 (2015)

    MathSciNet  MATH  Google Scholar 

  8. Breuil, C., Hellmann, E., Schraen, B.: Une interprétation modulaire de la variété trianguline. Math. Ann. 367(3–4), 1587–1645 (2017)

    MathSciNet  MATH  Google Scholar 

  9. Böckle, G., Juschka, A.-K.: Irreducibility of versal deformation rings in the \((p, p)\)-case for 2-dimensional representations. J. Algebra 444, 81–123 (2015)

    MathSciNet  MATH  Google Scholar 

  10. Barthel, L., Livné, R.: Irreducible modular representations of \( \text{ GL}_2\) of a local field. Duke Math. J. 75(2), 261–292 (1994)

    MathSciNet  MATH  Google Scholar 

  11. Barnet-Lamb, T., Gee, T., Geraghty, D., Taylor, R.: Potential automorphy and change of weight. Ann. Math. (2) 179(2), 501–609 (2014)

    MathSciNet  MATH  Google Scholar 

  12. 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)

    MathSciNet  MATH  Google Scholar 

  13. Boston, N., Lenstra Jr., H.W., Ribet, K.A.: Quotients of group rings arising from two-dimensional representations. C. R. Acad. Sci. Paris Sér. I Math. 312(4), 323–328 (1991)

    MathSciNet  MATH  Google Scholar 

  14. Breuil, C., Mézard, A.: Multiplicités modulaires et représentations de \( \text{ GL}_2({ Z}_p)\) et de \( \text{ Gal }(\overline{{ Q}}_p/{ Q}_p)\) en \(l=p\). Duke Math. J. 115(2), 205–310 (2002) (With an appendix by Guy Henniart)

  15. Breuil, C., Paškūnas, V.: Towards a modulo \(p\) Langlands correspondence for \( \text{ GL}_2\). Mem. Am. Math. Soc. 216(1016), vi+114 (2012)

    MATH  Google Scholar 

  16. Breuil, C.: Sur quelques représentations modulaires et \(p\)-adiques de \( \text{ GL}_2(\mathbb{Q}_p)\). I. Compos. Math. 138(2), 165–188 (2003)

    MathSciNet  MATH  Google Scholar 

  17. Calegari, F.: Even Galois representations and the Fontaine-Mazur conjecture. II. J. Am. Math. Soc. 25(2), 533–554 (2012)

    MathSciNet  MATH  Google Scholar 

  18. Carayol, H.: Sur les représentations \(l\)-adiques associées aux formes modulaires de Hilbert. Ann. Sci. École Norm. Sup. (4) 19(3), 409–468 (1986)

    MathSciNet  MATH  Google Scholar 

  19. Colmez, P., Dospinescu, G.: Complétés universels de représentations de \(\text{ GL}_2(\mathbb{Q}_p)\). Algebra Number Theory 8(6), 1447–1519 (2014)

    MathSciNet  MATH  Google Scholar 

  20. Colmez, P., Dospinescu, G., Paškūnas, V.: Irreducible components of deformation spaces: wild 2-adic exercises. Int. Math. Res. Not. IMRN 14, 5333–5356 (2015)

    MathSciNet  MATH  Google Scholar 

  21. Caraiani, A., Emerton, M., Gee, T., Geraghty, D., Paškūnas, V., Shin, S.W.: Patching and the \(p\)-adic local Langlands correspondence. Camb. J. Math. 4(2), 197–287 (2016)

    MathSciNet  MATH  Google Scholar 

  22. 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)

    MathSciNet  MATH  Google Scholar 

  23. Chenevier, G.: Sur la variété des caractéres p-adique du groupe de Galois absolu de \(\mathbb{Q}_p\) (2009). http://gaetan.chenevier.perso.math.cnrs.fr/articles/lieugalois.pdf

  24. Colmez, P.: Représentations de \( \text{ GL}_2(\mathbb{Q}_p)\) et \((\phi,\Gamma )\)-modules. Astérisque 330, 281–509 (2010)

    Google Scholar 

  25. Darmon, H.: Diamond, F., Taylor, R.: Fermat’s last theorem. In: Current developments in mathematics, 1995, pp. 1–154. Int. Press, Cambridge (1994)

  26. Ding, Y.: \(\cal{L}\)-invariants and local-global compatibility for the group \( \text{ GL}_2/F\). Forum Math. Sigma 4, e13, 49 (2016)

  27. Emerton, M., Gee, T.: A geometric perspective on the Breuil-Mézard conjecture. J. Inst. Math. Jussieu 13(1), 183–223 (2014)

    MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  29. Emerton, M.: A local-global compatibility conjecture in the \(p\)-adic Langlands programme for \( \text{ GL}_{2/{\mathbb{Q}}}\). Pure Appl. Math. Q. 2(2, Special Issue: In honor of John H. Coates. Part 2), 279–393 (2006)

  30. Emerton, M.: Ordinary parts of admissible representations of \(p\)-adic reductive groups I. Definition and first properties. Astérisque 331, 355–402 (2010)

    MathSciNet  MATH  Google Scholar 

  31. Emerton, M.: Ordinary parts of admissible representations of \(p\)-adic reductive groups II. Derived functors. Astérisque 331, 403–459 (2010)

    MathSciNet  MATH  Google Scholar 

  32. Emerton, M.: Local-global compatibility conjecture in the \(p\)-adic Langlands programme for \( \text{ GL}_{2/{\mathbb{Q}}}\) (2011). http://www.math.uchicago.edu/~emerton/pdffiles/lg.pdf

  33. Emerton, M., Paskunas, V.: On the density of supercuspidal points of fixed regular weight in local deformation rings and global Hecke algebras (2018). arXiv:1809.06598

  34. Fontaine, J.-M., Mazur, B.: Geometric Galois representations. Elliptic curves, modular forms, & Fermat’s last theorem (Hong Kong, 1993), Ser. Number Theory, vol. I, pp. 41–78. Int. Press, Cambridge (1995)

  35. Fontaine, J.-M.: Représentations \(\ell \)-adiques potentiellement semi-stables. Astérisque 223, 321–347 (1994)

  36. Geraghty, D.J.: Modularity lifting theorems for ordinary Galois representations. ProQuest LLC, Ann Arbor, MI, (2010). Thesis (Ph.D.)–Harvard University

  37. Gee, T., Kisin, M.: The Breuil-Mézard conjecture for potentially Barsotti–Tate representations. Forum Math. Pi 2:e1, 56 (2014)

  38. Gee, T., Newton, J.: Patching and the completed homology of locally symmetric spaces (2016). arXiv:1609.06965

  39. Gouvêa, F.Q.: Deformations of Galois representations. In: Arithmetic algebraic geometry (Park City, UT, 1999), volume 9 of IAS/Park City Math. Ser., pp. 233–406. American Mathematical Society, Providence, RI (2001) (Appendix 1 by Mark Dickinson, Appendix 2 by Tom Weston and Appendix 3 by Matthew Emerton)

  40. Hida, H.: Nearly ordinary Hecke algebras and Galois representations of several variables. Algebraic analysis. geometry, and number theory (Baltimore, MD, 1988), pp. 115–134. Johns Hopkins University Press, Baltimore (1989)

  41. Hida, H.: On nearly ordinary Hecke algebras for \( \text{ GL }(2)\) over totally real fields. In: Algebraic number theory, vol. 17 of Adv. Stud. Pure Math., pp. 139–169. Academic Press, Boston (1989)

  42. Hu, Y., Paškūnas, V.: On crystabelline deformation rings of \(\text{ Gal }(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)\) (with an appendix by Jack Shotton). Math. Ann. 373(1–2), 421–487 (2019)

    MathSciNet  MATH  Google Scholar 

  43. Harris, M., Taylor, R.: The geometry and cohomology of some simple Shimura varieties, vol. 151 of Annals of Mathematics Studies. Princeton University Press, Princeton (2001) (With an appendix by Vladimir G. Berkovich)

  44. Yongquan, H., Tan, F.: The Breuil–Mézard conjecture for non-scalar split residual representations. Ann. Sci. Éc. Norm. Supér. (4) 48(6), 1383–1421 (2015)

    MathSciNet  MATH  Google Scholar 

  45. Kisin, M.: Potentially semi-stable deformation rings. J. Am. Math. Soc. 21(2), 513–546 (2008)

    MathSciNet  MATH  Google Scholar 

  46. Kisin, M.: The Fontaine-Mazur conjecture for \( \text{ GL}_2\). J. Am. Math. Soc. 22(3), 641–690 (2009)

    MATH  Google Scholar 

  47. Kisin, M.: Modularity of 2-adic Barsotti-Tate representations. Invent. Math. 178(3), 587–634 (2009)

    MathSciNet  MATH  Google Scholar 

  48. Kisin, M.: Moduli of finite flat group schemes, and modularity. Ann. Math. (2) 170(3), 1085–1180 (2009)

    MathSciNet  MATH  Google Scholar 

  49. Khare, C.B., Thorne, J.A.: Potential automorphy and the Leopoldt conjecture. Am. J. Math. 139(5), 1205–1273 (2017)

    MathSciNet  MATH  Google Scholar 

  50. 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. (2) 169(1), 229–253 (2009)

    MathSciNet  MATH  Google Scholar 

  51. Khare, C., Wintenberger, J.-P.: Serre’s modularity conjecture. II. Invent. Math. 178(3), 505–586 (2009)

    MathSciNet  MATH  Google Scholar 

  52. Pan, L.: The Fontaine-Mazur conjecture in the residually reducible case (2019). arXiv:1901.07166

  53. Paškūnas, V.: Extensions for supersingular representations of \( \text{ GL}_2(\mathbb{Q}_p)\). Astérisque 331, 317–353 (2010)

  54. Paškūnas, V.: The image of Colmez’s Montreal functor. Publ. Math. Inst. Hautes Études Sci. 118, 1–191 (2013)

    MathSciNet  MATH  Google Scholar 

  55. Paškūnas, V.: Blocks for \( \text{ mod }\,p\) representations of \( \text{ GL}_2(\mathbb{Q}_p)\). In: Automorphic forms and Galois representations, vol. 2, volume 415 of London Math. Soc. Lecture Note Ser., pp. 231–247. Cambridge University Press, Cambridge (2014)

  56. Paškūnas, V.: On the Breuil–Mézard conjecture. Duke Math. J. 164(2), 297–359 (2015)

    MathSciNet  MATH  Google Scholar 

  57. Paškūnas, V.: On 2-dimensional 2-adic Galois representations of local and global fields. Algebra Number Theory 10(6), 1301–1358 (2016)

    MathSciNet  MATH  Google Scholar 

  58. Paškūnas, V.: On 2-adic deformations. Math. Z. 286(3–4), 801–819 (2017)

    MathSciNet  MATH  Google Scholar 

  59. Pyvovarov, A.: On the Breuil–Schneider conjecture: Generic case (2018). arXiv:1803.01610

  60. Saito, T.: Hilbert modular forms and \(p\)-adic Hodge theory. Compos. Math. 145(5), 1081–1113 (2009)

    MathSciNet  MATH  Google Scholar 

  61. Sasaki, S.: Integral models of Hilbert modular varieties in the ramified case, deformations of modular Galois representations, and weight one forms II (2017). http://www.cantabgold.net/users/s.sasaki.03/hmv1-5-9.pdf

  62. Sasaki, S.: Integral models of Hilbert modular varieties in the ramified case, deformations of modular Galois representations, and weight one forms. Invent. Math. 215(1), 171–264 (2019)

    MathSciNet  MATH  Google Scholar 

  63. Shotton, J.: Local deformation rings for \( \text{ GL}_2\) and a Breuil–Mézard conjecture when \(\ell \ne p\). Algebra Number Theory 10(7), 1437–1475 (2016)

    MathSciNet  MATH  Google Scholar 

  64. Snowden, A.: On two dimensional weight two odd representations of totally real fields (2009). arXiv:0905.4266

  65. Taylor, R.: On Galois representations associated to Hilbert modular forms. Invent. Math. 98(2), 265–280 (1989)

    MathSciNet  MATH  Google Scholar 

  66. Taylor, R.: On the meromorphic continuation of degree two \(L\)-functions. Doc. Math., (Extra Vol.), pp. 729–779 (2006)

  67. Taylor, R.: Automorphy for some \(l\)-adic lifts of automorphic mod \(l\) Galois representations. II. Publ. Math. Inst. Hautes Études Sci. 108, 183–239 (2008)

  68. Thorne, J.: On the automorphy of \(l\)-adic Galois representations with small residual image. J. Inst. Math. Jussieu 11(4), 855–920 (2012) (With an appendix by Robert Guralnick, Florian Herzig, Richard Taylor and Thorne)

  69. Thorne, J.A.: Automorphy lifting for residually reducible \(l\)-adic Galois representations. J. Am. Math. Soc. 28(3), 785–870 (2015)

    MathSciNet  MATH  Google Scholar 

  70. Tung, S.-N.: On the automorphy of 2-dimensional potentially semi-stable deformation rings of \(\text{ Gal }(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)\) (2018). arXiv:1803.07451

  71. Vignéras, M.-F.: Representations modulo \(p\) of the \(p\)-adic group \( \text{ GL }(2, F)\). Compos. Math. 140(2), 333–358 (2004)

    MathSciNet  MATH  Google Scholar 

  72. Wiles, A.: On ordinary \(\lambda \)-adic representations associated to modular forms. Invent. Math. 94(3), 529–573 (1988)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

I would like to thank my advisor Vytautas Paškūnas, for suggesting me to work on this project and sharing with his profound insight and ideas. I also thanks Shu Sasaki for many helpful discussions on modularity lifting theorems and for pointing out many inaccuracies in an earlier draft, and Jack Thorne for his hospitality during my visit to Cambridge in May 2018 and for answering my questions regarding 2-adic modularity lifting theorems. I would also like to thank Patrick Allen and the anonymous referee for many useful suggestions, comments, and corrections. This research was funded in part by the DFG, SFB/TR 45 “Periods, moduli spaces and arithmetic of algebraic varieties”.

Author information

Authors and Affiliations

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

    Shen-Ning Tung

Authors
  1. Shen-Ning Tung
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shen-Ning Tung.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tung, SN. On the modularity of 2-adic potentially semi-stable deformation rings. Math. Z. 298, 107–159 (2021). https://doi.org/10.1007/s00209-020-02588-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-020-02588-4

Search

Navigation