Mathematische Zeitschrift

, Volume 288, Issue 1–2, pp 199–215 | Cite as

Constructing Hilbert modular forms without exceptional primes



In this paper we construct families of Hilbert modular newforms without exceptional primes. This is achieved by generalizing the notion of good-dihedral primes, introduced by Khare and Wintenberger in their proof of Serre’s modularity conjecture, to totally real fields.


Galois representations Hilbert modular forms Level raising theorem Inertial types 

Mathematics Subject Classification

11F80 11F41 



This paper is part of the second author’s Ph.D. thesis. Part of this work has been written during a stay at the Hausdorff Research Institute for Mathematics and a stay, of the second author, at the Mathematical Institute of the University of Barcelona. The authors would like to thank these institutions for their support and the optimal working conditions. The research of L.V.D. was supported by an ICREA Academia Research Prize and by MICINN grant MTM2012-33830. The research of A.Z. was supported by the CONACYT Grant No. 432521/286915.


  1. 1.
    Arias-de-Reyna, S., Dieulefait, L., Wiese, G.: Compatible systems of symplectic Galois representations and the inverse Galois problem I. Images of projective representations. Trans. Amer. Math. Soc. 369(2), 887–908 (2017)Google Scholar
  2. 2.
    Bockle, G.: Deformation of Galois representations. In: Darmon, H., Diamond, F., Dieulefait, L.V., Edixhoven, B., Rotger, V. (eds.) Elliptic Curves, Hilbert Modular Forms and Galois Deformations. Progress in Math., pp. 21–115. Birkhauser, Basel (2013)CrossRefGoogle Scholar
  3. 3.
    Breuil, C.: Une remarque sur les reprèsentations locales \(p\)-adiques et les congruences entre formes modulaires de Hilbert. Bull. Soc. Math. Fr. 127, 459–472 (1999)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Carayol, H.: Sur les reprèsentations \(\ell \)-adiques associées aux formes modulaires de Hilbert. Ann. Sci. École Norm. Sup. Sér. 4 19(3), 409–468 (1986)CrossRefMATHGoogle Scholar
  5. 5.
    Dieulefait, L., Wiese, G.: On modular forms and the inverse Galois problem. Trans. Am. Math. Soc. 363(9), 4569–4584 (2011)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Dimitrov, M.: Valeur critique de la fonction L adjointe d’une forme modulaire de Hilbert et arithmétique du motif correspondant. Ph.D. thesis, Université Paris 13 (2003)Google Scholar
  7. 7.
    Dimitrov, M.: Galois representations modulo \(p\) and cohomology of Hilbert modular varietes. Ann. Sci. École Norm. Sup. Sér. 4 38(4), 505–551 (2005)CrossRefMATHGoogle Scholar
  8. 8.
    Gee, T.: Automorphic lifts of prescribed types. Math. Ann. 350(1), 107–144 (2011)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Gee, T.: Modularity lifting theorems. Arizona winter school. (2013). Accessed Nov 2013
  10. 10.
    Henniart, G.: Sur l’unicité des types pour GL(2). Duke Math. J. 155(2), 298–310 (2002)Google Scholar
  11. 11.
    Jarvis, F.: Level lowering for modular mod \(l\) Galois representations over totally real fields. Math. Ann. 313(1), 141–160 (1999)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Khare, C., Wintenberger, J.P.: Serre’s modularity conjecture I. Invent. Math. 178(3), 485–504 (2009)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Khare, C., Wintenberger, J.P.: Serre’s modularity conjecture, II. Invent. Math. 178(3), 505–586 (2009)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Shimura, G.: The special values of the zeta functions associated with Hilbert modular forms. Duke Math. J. 45(3), 637–679 (1978)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Tate, J.: Number theoretic background. Proc. Symp. Pure Math. 33(part 2), 3–26 (1979)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Taylor, R.: On Galois representations associated to Hilbert modular forms. Invent. Math. 98(2), 265–280 (1989)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Taylor, R.: On Galois representations associated to Hilbert modular forms II. In: Coates, J., Yau, S.T. (eds.) Elliptic Curves, Modular Forms and Fermat’s Last Theorem, 2nd edn, pp. 333–340. International Press, Boston (1997)Google Scholar
  18. 18.
    Taylor, R.: On icosahedral Artin representations. Am. J. Math. 125(3), 549–566 (2003)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Weinstein, J.: Hilbert modular forms with prescribed ramification. Int. Math. Res. Not. 8, 1388–1420 (2009)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Wiese, G.: On projective linear groups over finite fields as Galois groups over the rational numbers. In: van der Geer, G., Endixhoven, B., Moonen, B. (eds.) Modular Forms on Schiermonnikoog, pp. 343–350. Cambridge University Press, Cambridge (2008)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Departament d’Algebra i Geometria, Facultat de MatemàtiquesUniversitat de BarcelonaBarcelonaSpain
  2. 2.Instituto de Matemáticas, (Unidad Cuernavaca) UNAMCuernavacaMexico

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