Computing Automorphic Forms on Shimura Curves over Fields with Arbitrary Class Number

  • John Voight
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6197)


We extend methods of Greenberg and the author to compute in the cohomology of a Shimura curve defined over a totally real field with arbitrary class number. Via the Jacquet-Langlands correspondence, we thereby compute systems of Hecke eigenvalues associated to Hilbert modular forms of arbitrary level over a totally real field of odd degree. We conclude with two examples which illustrate the effectiveness of our algorithms.


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  1. 1.
    Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symbolic Comput. 24(3-4), 235–265 (1997)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Cremona, J.: The elliptic curve database for conductors to 130000. In: Hess, F., Pauli, S., Pohst, M. (eds.) ANTS 2006. LNCS, vol. 4076, pp. 11–29. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  3. 3.
    Deligne, P.: Travaux de Shimura. Séminaire Bourbaki, Lecture notes in Math. 244(389), 123–165Google Scholar
  4. 4.
    Dembélé, L.: Explicit computations of Hilbert modular forms on \(\mathbb{Q}(\sqrt{5})\). Experiment. Math. 14(4), 457–466 (2005)MATHMathSciNetGoogle Scholar
  5. 5.
    Dembélé, L.: Quaternionic Manin symbols, Brandt matrices and Hilbert modular forms. Math. Comp. 76(258), 1039–1057 (2007)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Dembélé, L., Donnelly, S.: Computing Hilbert modular forms over fields with nontrivial class group. In: van der Poorten, A.J., Stein, A. (eds.) ANTS-VIII 2008. LNCS, vol. 5011, pp. 371–386. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  7. 7.
    Donnelly, S., Voight, J.: Tables of Hilbert modular forms and elliptic curves over totally real fields (in preparation)Google Scholar
  8. 8.
    Greenberg, M., Voight, J.: Computing systems of Hecke eigenvalues associated to Hilbert modular forms. Math. Comp. (accepted)Google Scholar
  9. 9.
    Gunnells, P., Yasaki, D.: Hecke operators and Hilbert modular forms. In: van der Poorten, A.J., Stein, A. (eds.) ANTS-VIII 2008. LNCS, vol. 5011, pp. 387–401. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  10. 10.
    Hida, H.: On abelian varieties with complex multiplication as factors of the Jacobians of Shimura curves. American Journal of Mathematics 103(4), 727–776 (1981)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Hida, H.: Hilbert modular forms and Iwasawa theory. Clarendon Press, Oxford (2006)MATHCrossRefGoogle Scholar
  12. 12.
    Kirschmer, M., Voight, J.: Algorithmic enumeration of ideal classes for quaternion orders. SIAM J. Comput. (SICOMP) 39(5), 1714–1747 (2010)CrossRefMathSciNetGoogle Scholar
  13. 13.
    Stein, W.A.: Modular forms database (2004),
  14. 14.
    Stein, W.A., Watkins, M.: A database of elliptic curves—first report. In: Fieker, C., Kohel, D.R. (eds.) ANTS 2002. LNCS, vol. 2369, pp. 267–275. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  15. 15.
    Vignéras, M.-F.: Arithmétique des algèbres de quaternions. LNM, vol. 800. Springer, Berlin (1980)MATHGoogle Scholar
  16. 16.
    Voight, J.: Computing fundamental domains for cofinite Fuchsian groups. J. Théorie Nombres Bordeaux 21(2), 467–489 (2009)MathSciNetGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • John Voight
    • 1
  1. 1.Department of Mathematics and StatisticsUniversity of VermontBurlingtonUSA

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