Computing Power Series Expansions of Modular Forms

  • John VoightEmail author
  • John Willis
Part of the Contributions in Mathematical and Computational Sciences book series (CMCS, volume 6)


We exhibit a method to numerically compute power series expansions of modular forms on a cocompact Fuchsian group, using the explicit computation of a fundamental domain and linear algebra. As applications, we compute Shimura curve parametrizations of elliptic curves over a totally real field, including the image of CM points, and equations for Shimura curves.



The authors would like to thank Srinath Baba, Valentin Blomer, Noam Elkies, David Gruenewald, Paul Nelson, Kartik Prasanna, Victor Rotger, and Frederik Strömberg for helpful comments on this work. The authors were supported by NSF grant DMS-0901971.


  1. [BG12]
    S. Baba, H. Granath, Differential equations and expansions for quaternionic modular forms in the discriminant 6 case. LMS J. Comput. Math. 15, 385–399 (2012) CrossRefMathSciNetGoogle Scholar
  2. [Bay02]
    P. Bayer, Uniformization of certain Shimura curves. Differ. Galois Theory 58, 13–26 (2002) CrossRefMathSciNetGoogle Scholar
  3. [BT07]
    P. Bayer, A. Travesa, Uniformization of triangle modular curves. Publ. Mat. 51, 43–106 (2007) CrossRefzbMATHMathSciNetGoogle Scholar
  4. [BT08]
    P. Bayer, A. Travesa, On local constants associated to arithmetical automorphic functions. Pure Appl. Math. Q. 4(4), 1107–1132 (2008) CrossRefzbMATHMathSciNetGoogle Scholar
  5. [Bea95]
    A. Beardon, The Geometry of Discrete Groups. Grad. Texts in Math., vol. 91 (Springer, New York, 1995) Google Scholar
  6. [BDP13]
    M. Bertolini, H. Darmon, K. Prasanna, Generalized Heegner cycles and p-adic Rankin L-series. Duke Math. J. 162(6), 1033–1148 (2013), with an appendix by Brian Conrad CrossRefzbMATHMathSciNetGoogle Scholar
  7. [BJTX12]
    J. Beyerl, K. James, C. Trentacose, H. Xue, Products of nearly holomorphic eigenforms. Ramanujan J. 27(3), 377–386 (2012) CrossRefzbMATHMathSciNetGoogle Scholar
  8. [Ste12]
    J. Bober, A. Deines, A. Klages-Mundt, B. LeVeque, R.A. Ohana, A. Rabindranath, P. Sharaba, W. Stein, A database of elliptic curves over \(\mathbb {Q}(\sqrt{5} )\)—first report, to appear in ANTS X: Proceedings of the Tenth Algorithmic Number Theory Symposium, 2012 Google Scholar
  9. [BSV06]
    A.R. Booker, A. Strömbergsson, A. Venkatesh, Effective computation of Maass cusp forms. Int. Math. Res. Not., posted on 2006, Art. ID 71281, 34 Google Scholar
  10. [BJP97]
    W. Bosma, J. Cannon, C. Playoust, The Magma algebra system. I. The user language. J. Symb. Comput. 24(3–4), 235–265 (1997) CrossRefzbMATHMathSciNetGoogle Scholar
  11. [Car86]
    H. Carayol, Sur la mauvaise réduction des courbes de Shimura. Compos. Math. 59, 151–230 (1986) zbMATHMathSciNetGoogle Scholar
  12. [CQ05]
    G. Cardona, J. Quer, Field of moduli and field of definition for curves of genus 2, in Computational Aspects of Algebraic Curves, ed. by T. Shaska. Lecture Notes Series on Computing, vol. 13 (2005), pp. 71–83 CrossRefGoogle Scholar
  13. [Coh93]
    H. Cohen, A Course in Computational Algebraic Number Theory. Grad. Texts in Math., vol. 138 (Springer, New York, 1993) CrossRefzbMATHGoogle Scholar
  14. [DG08]
    B. Datskovsky, P. Guerzhoy, p-adic interpolation of Taylor coefficients of modular forms. Math. Ann. 340(2), 465–476 (2008) CrossRefzbMATHMathSciNetGoogle Scholar
  15. [Dem05]
    L. Dembélé, Explicit computations of Hilbert modular forms on \(\mathbb {Q}(\sqrt{5} )\). Exp. Math. 14(4), 457–466 (2005) CrossRefzbMATHGoogle Scholar
  16. [DV]
    L. Dembélé, J. Voight, Explicit methods for Hilbert modular forms. Accepted to “Elliptic Curves, Hilbert Modular Forms and Galois Deformations” Google Scholar
  17. [DN67]
    K. Doi, H. Naganuma, On the algebraic curves uniformized by arithmetical automorphic functions. Ann. Math. 86(3), 449–460 (1967) CrossRefzbMATHMathSciNetGoogle Scholar
  18. [Elk98]
    N.D. Elkies, Shimura curve computations, in Algorithmic Number Theory, Portland, OR, 1998. Lecture Notes in Comput. Sci., vol. 1423 (Springer, Berlin, 1998), pp. 1–47 CrossRefGoogle Scholar
  19. [Fra11]
    C. Franc, Nearly rigid analytic modular forms and their values at CM points. PhD. thesis, McGill University, 2011 Google Scholar
  20. [Gal96]
    S.D. Galbraith, Equations for modular curves. PhD. thesis, University of Oxford, 1996 Google Scholar
  21. [GV11]
    M. Greenberg, J. Voight, Computing systems of Hecke eigenvalues associated to Hilbert modular forms. Math. Comput. 80, 1071–1092 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  22. [Hej92]
    D.A. Hejhal, Eigenvalues of the Laplacian for Hecke triangle groups. Mem. Amer. Math. Soc. 97 (1992), no. 469 Google Scholar
  23. [Hej99]
    D.A. Hejhal, On eigenfunctions of the Laplacian for Hecke triangle groups, in Emerging Applications of Number Theory, ed. by D. Hejhal, J. Friedman et al. IMA Vol. Math. Appl., vol. 109 (Springer, New York, 1999), pp. 291–315 CrossRefGoogle Scholar
  24. [Hej12]
    D.A. Hejhal, On the calculation of Maass cusp forms, in Hyperbolic Geometry and Applications in Quantum Chaos and Cosmology, ed. by J. Bolte, London Math. Soc. Lecture Note Series, vol. 397, F. Steiner (Cambridge University Press, Cambridge, 2012), pp. 175–185 Google Scholar
  25. [Kat92]
    S. Katok, Fuchsian Groups (University of Chicago Press, Chicago, 1992) zbMATHGoogle Scholar
  26. [LLL82]
    A.K. Lenstra, H.W. Lenstra, L. Lovász, Factoring polynomials with rational coefficients. Math. Ann. 261, 513–534 (1982) CrossRefGoogle Scholar
  27. [Mes91]
    J.-F. Mestre, Construction de courbes de genre 2 a partir de leurs modules, in Effective Methods in Algebraic Geometry, Castiglioncello, 1990 (Birkhäuser, Boston, 1991), pp. 313–334 CrossRefGoogle Scholar
  28. [Mor83]
    C.J. Moreno, The Chowla-Selberg formula. J. Number Theory 17, 226–245 (1983) CrossRefzbMATHMathSciNetGoogle Scholar
  29. [Mori95]
    A. Mori, Power series expansions of modular forms at CM points. Rend. Semin. Mat. Univ. Politec. Torino 53(4), 361–374 (1995) zbMATHGoogle Scholar
  30. [Mori11]
    A. Mori, Power series expansions of modular forms and their interpolation properties. Int. J. Number Theory 7(2), 529–577 (2011) CrossRefzbMATHMathSciNetGoogle Scholar
  31. [Nel11]
    P. Nelson, Computing on Shimura curves, Appendices B and C of Periods and special values of L-functions, notes from Arizona Winter School 2011.
  32. [Nel12]
    P.D. Nelson, Evaluating modular forms on Shimura curves (2012). arXiv:1210.1243
  33. [O’SR13]
    C. O’Sullivan, M.S. Risager, Non-vanishing of Taylor coefficients, Poincaré series and central values of L-functions. Ramanujan J. 30(1), 67–100 (2013) CrossRefzbMATHMathSciNetGoogle Scholar
  34. [RZ93]
    F. Rodriguez-Villegas, D. Zagier, Square roots of central values of Hecke L-series, in Advances in Number Theory, Kingston, ON, 1991. Oxford Sci. Publ. (Oxford Univ. Press, New York, 1993), pp. 81–99 Google Scholar
  35. [Shi67]
    G. Shimura, Construction of class fields and zeta functions of algebraic curves. Ann. Math. 85, 58–159 (1967) CrossRefzbMATHMathSciNetGoogle Scholar
  36. [Shi75]
    G. Shimura, On some arithmetic properties of modular forms of one and several variables. Ann. Math. 102, 491–515 (1975) CrossRefzbMATHMathSciNetGoogle Scholar
  37. [Shi77]
    G. Shimura, On the derivatives of theta functions and modular forms. Duke Math. J. 44, 365–387 (1977) CrossRefzbMATHMathSciNetGoogle Scholar
  38. [Shi79]
    G. Shimura, Automorphic forms and the periods of abelian varieties. J. Math. Soc. Jpn. 31(3), 561–592 (1979) CrossRefzbMATHMathSciNetGoogle Scholar
  39. [Shi02]
    G. Shimura, Collected Papers, vol. I–IV (Springer, New York, 2002–2003) CrossRefGoogle Scholar
  40. [Sta84]
    H.M. Stark, Fourier coefficients of Maass waveforms, in Modular Forms, Durham, 1983. Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res. (Horwood, Chichester, 1984), pp. 263–269 Google Scholar
  41. [The05]
    H. Then, Maass cusp forms for large eigenvalues. Math. Comput. 74(249), 363–381 (2005) CrossRefzbMATHMathSciNetGoogle Scholar
  42. [Vig80]
    M.-F. Vignéras, Arithmétique des algèbres de quaternions. Lect. Notes in Math., vol. 800 (Springer, Berlin, 1980) zbMATHGoogle Scholar
  43. [Voi05]
    J. Voight, Quadratic forms and quaternion algebras: algorithms and arithmetic. PhD. thesis, University of California, Berkeley, 2005 Google Scholar
  44. [Voi09]
    J. Voight, Computing fundamental domains for Fuchsian groups. J. Théor. Nr. Bordx. 21(2), 467–489 (2009) CrossRefMathSciNetGoogle Scholar
  45. [Voi]
    J. Voight, Identifying the matrix ring: algorithms for quaternion algebras and quadratic forms, in Quadratic and Higher Degree Forms. Developments in Math., vol. 31 (Springer, New York, 2013), pp. 255–298 CrossRefGoogle Scholar
  46. [Zag08]
    D. Zagier, Elliptic Modular Forms and Their Applications, the 1-2-3 of Modular Forms: Lectures at a Summer School in Nordfjordeid, Norway, Universitext (Springer, Berlin, 2008), pp. 1–103 Google Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsUniversity of VermontBurlingtonUSA
  2. 2.Department of MathematicsDartmouth CollegeHanoverUSA
  3. 3.Department of MathematicsThe University of Colorado at BoulderBoulderUSA

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