Inventiones mathematicae

, Volume 184, Issue 1, pp 1–46

On the weights of mod p Hilbert modular forms

Article

Abstract

We prove many cases of a conjecture of Buzzard, Diamond and Jarvis on the possible weights of mod p Hilbert modular forms, by making use of modularity lifting theorems and computations in p-adic Hodge theory.

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References

  1. 1.
    Ash, A., Stevens, G.: Modular forms in characteristic l and special values of their L-functions. Duke Math. J. 53(3), 849–868 (1986) CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Breuil, C., Conrad, B., Diamond, F., Taylor, R.: On the modularity of elliptic curves over Q: wild 3-adic exercises. J. Am. Math. Soc. 14(4), 843–939 (2001) (electronic) CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Buzzard, K., Diamond, F., Jarvis, F.: On Serre’s conjecture for mod l Galois representations over totally real fields. Duke Math. J. (2010, to appear) Google Scholar
  4. 4.
    Breuil, C., Mézard, A.: 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(2), 205–310 (2002). With an appendix by Guy Henniart CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Breuil, C.: Construction de représentations p-adiques semi-stables. Ann. Sci. École Norm. Super. (4) 31(3), 281–327 (1998) MATHMathSciNetGoogle Scholar
  6. 6.
    Breuil, C.: Représentations semi-stables et modules fortement divisibles. Invent. Math. 136(1), 89–122 (1999) CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Breuil, C.: Groupes p-divisibles, groupes finis et modules filtrés. Ann. Math. (2) 152(2), 489–549 (2000) CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Caruso, X.: \(\mathbb{F}_{p}\)-représentations semi-stables. Preprint Google Scholar
  9. 9.
    Conrad, B., Diamond, F., Taylor, R.: Modularity of certain potentially Barsotti-Tate Galois representations. J. Am. Math. Soc. 12(2), 521–567 (1999) CrossRefMATHMathSciNetGoogle Scholar
  10. 10.
    Caruso, X., Liu, T.: Quasi-semi-stable representations. Bull. Soc. Math. Fr. 137(2), 185–223 (2009) MATHMathSciNetGoogle Scholar
  11. 11.
    Diamond, F.: A correspondence between representations of local Galois groups and Lie-type groups. In: L-functions and Galois representations. London Math. Soc. Lecture Note Ser., vol. 320, pp. 187–206. Cambridge University Press, Cambridge (2007) CrossRefGoogle Scholar
  12. 12.
    Fontaine, J.-M., Laffaille, G.: Construction de représentations p-adiques. Ann. Sci. École Norm. Super. (4) 15(4), 547–608 (1982) MATHMathSciNetGoogle Scholar
  13. 13.
    Gee, T.: A modularity lifting theorem for weight two Hilbert modular forms. Math. Res. Lett. 13(5–6), 805–811 (2006) MATHMathSciNetGoogle Scholar
  14. 14.
    Gee, T.: Companion forms over totally real fields, II. Duke Math. J. 136(2), 275–284 (2007) CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Gee, T.: Automorphic lifts of prescribed types. Math. Ann. (2010, to appear) Google Scholar
  16. 16.
    Gee, T., Savitt, D.: Serre weights for quaternion algebras. Compositio (2010, to appear) Google Scholar
  17. 17.
    Kisin, M.: Moduli of finite flat group schemes, and modularity. Ann. Math. (2008, to appear) Google Scholar
  18. 18.
    Kisin, M.: Potentially semi-stable deformation rings. J. Am. Math. Soc. 21(2), 513–546 (2008) CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Kisin, M.: Modularity of 2-adic Barsotti-Tate representations. Invent. Math. 178(3), 587–634 (2009) CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Savitt, D.: On a conjecture of Conrad, Diamond, and Taylor. Duke Math. J. 128(1), 141–197 (2005) CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    Savitt, D.: Breuil modules for Raynaud schemes. J. Number Theory 128(11), 2939–2950 (2008) CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Schein, M.: Weights of Galois representations associated to Hilbert modular forms. J. Reine Angew. Math. 622, 57–94 (2008) CrossRefMATHMathSciNetGoogle Scholar
  23. 23.
    Taylor, R.: On the meromorphic continuation of degree two L-functions. Doc. Math. 729–779 (2006) (electronic) Google Scholar

Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  1. 1.Department of MathematicsNorthwestern UniversityEvanstonUSA

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