The Hyperbolic Laplace-Beltrami Operator

  • Markus Szymon Fraczek
Part of the Lecture Notes in Mathematics book series (LNM, volume 2139)


In this chapter we will introduce some basic concepts of hyperbolic geometry and automorphic forms. A variety of books is available which provide a more comprehensive description of the relevant material. Hejhal’s books about the Selberg trace formula [58] and [59] are a source of exhaustive informations regarding most topics discussed in this chapter, these books are most useful for researches already familiar with most of the concepts. Iwaniec’s book [68] is more introductory in nature, discussing the relevant subjects in an accessible way. Bump’s book [25] covers both the classical and the representation theoretic views of automorphic forms. Bruggeman’s book on families of automorphic forms [21] is especially relevant in regard of deformations of automorphic forms, discussing their dependency on the weight and the character. For introductory articles on the spectral theory on hyperbolic surfaces and the Selberg trace formula see [14] and [83].


Fundamental Domain Eisenstein Series Cusp Form Trace Formula Automorphic Form 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 2.
    Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions. Dover, New York (1964)zbMATHGoogle Scholar
  2. 6.
    Avelin, H.: Deformation of Γ 0(5)-cusp forms. Math. Comput. 76, 361–384 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 7.
    Avelin, H.: Computations of Eisenstein series on Fuchsian groups. Math. Comput. 77, 1779–1800 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 13.
    Booker, A., Strömbergsson, A., Venkatesh, A.: Effective computation of Maass cusp forms. IMRN 2006, 1–34 (2006)zbMATHGoogle Scholar
  5. 14.
    Borthwick, D.: Introduction to spectral theory on hyperbolic surfaces. Proc. Symp. Pure Math. 84, 3–48 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 21.
    Bruggeman, R.: Families of Automorphic Forms. Birkhäuser, Basel/Boston (1994)CrossRefzbMATHGoogle Scholar
  7. 23.
    Bruggeman, R., Lewis, J., Zagier, D.: Function theory related to the group \(\mathrm{PSL}\!\left (2, \mathbb{R}\right )\). Dev. Math. 28, 107–201 (2013)MathSciNetzbMATHGoogle Scholar
  8. 24.
    Bruggeman, R., Lewis, J., Zagier, D.: Period Functions for Maass Forms and Cohomology. Memoirs of the American Mathematical Society. American Mathematical Society, Providence (2015)zbMATHGoogle Scholar
  9. 25.
    Bump, D.: Automorphic Forms and Representations. Cambridge University Press, Cambridge/New York (1998)zbMATHGoogle Scholar
  10. 29.
    Chang, C.H., Mayer, D.: The transfer operator approach to Selberg’s zeta function and modular and Maass wave forms for \(\mathrm{PSL}\!\left (2, \mathbb{Z}\right )\). In: Hejhal, D., Gutzwiller, M., et al. (eds.) Emerging Applications of Number Theory, pp. 72–142. Springer, New York (1999)Google Scholar
  11. 32.
    Deitmar, A., Hilgert, J.: A Lewis correspondence for submodular groups. Forum Mathematicum 19 (6), 1075–1099 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 42.
    Fraczek, M.: Spezielle Eigenfunktionen des Transfer-Operators für Hecke Kongruenz Untergruppen. Diploma Thesis, Clausthal University (2006)Google Scholar
  13. 44.
    Fraczek, M., Mayer, D.: Symmetries of the transfer operator for Γ 0(n) and a character deformation of the Selberg zeta function for Γ 0(4). Algebra Number Theory 6 (3), 587–610 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 45.
    Fraczek, M., Mayer, D., Mühlenbruch, T.: A realization of the Hecke algebra on the space of period functions for Γ 0(n). J. reine angew. Math. 603, 133–163 (2007)MathSciNetzbMATHGoogle Scholar
  15. 56.
    Guillope, L., Lin, K., Zworski, M.: The Selberg zeta function for convex co-compact Schottky groups. Commun. Math. Phys. 245 (1), 149–175 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 58.
    Hejhal, D.: The Selberg Trace Formula for \(\mathrm{PSL}\!\left (2, \mathbb{R}\right )\), Volume 1. Lecture Notes in Mathematics, vol. 548. Springer, Berlin/Heidelberg (1976)Google Scholar
  17. 59.
    Hejhal, D.: The Selberg Trace Formula for \(\mathrm{PSL}\!\left (2, \mathbb{R}\right )\), Volume 2. Lecture Notes in Mathematics, vol. 1001. Springer, Berlin/Heidelberg (1983)Google Scholar
  18. 61.
    Hejhal, D.: On the calculation of Maass cusp forms. In: Bolte, J., Steiner, F. (eds.) Hyperbolic Geometry and Applications in Quantum Chaos and Cosmology, pp. 175–186. Cambridge University Press, Cambridge, UK (2011)CrossRefGoogle Scholar
  19. 63.
    Hilgert, J., Mayer, D., Movasati, H.: Transfer operators for Γ 0(n) and the Hecke operators for period functions of \(\mathrm{PSL}\!\left (2, \mathbb{Z}\right )\). Math. Proc. Camb. Philos. Soc. 139, 81–116 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 66.
    Huxley, M.: Scattering matrices for congruence subgroups. In: Modular Forms (Durham, 1983). Ellis Horwood Series in Mathematics and Its Applications: Statistics and Operational Research, pp. 141–156. Horwood, Chichester (1984)Google Scholar
  21. 68.
    Iwaniec, H.: Spectral Methods of Automorphic Forms. American Mathematical Society, Providence (2002)CrossRefzbMATHGoogle Scholar
  22. 78.
    Lewis, J., Zagier, D.: Period functions for Maass wave forms, I. Ann. Math. 153, 191–258 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 83.
    Marklof, J.: Selberg’s trace formula: an introduction. In: Bolte, J., Steiner, F. (eds.) Hyperbolic Geometry and Applications in Quantum Chaos and Cosmology, pp. 83–120. Cambridge University Press, Cambridge, UK (2011)CrossRefGoogle Scholar
  24. 84.
    Matthies, C., Steiner, F.: Selberg’s ζ function and the quantization of chaos. Phys. Rev. A 44 (12), R7877–R7880 (1991)MathSciNetCrossRefGoogle Scholar
  25. 95.
    Miyake, T.: Modular Forms. Springer Monographs in Mathematics. Springer, Berlin/New York (2006)zbMATHGoogle Scholar
  26. 99.
    Mühlenbruch, T.: Hecke operators on period functions for Γ 0(n). J. Number Theory 118, 208–235 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 103.
    Phillips, R., Sarnak, P.: The spectrum of fermat curves. Geom. Funct. Anal. 1, 80–146 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 104.
    Phillips, R., Sarnak, P.: Cusp forms for character varieties. Geom. Funct. Anal. 4, 93–118 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 119.
    Selberg, A.: Remarks on the distribution of poles of Eisenstein series. In: Festschrift in Honor of I.I. Piatetski-Shapiro, vol. 2, pp. 251–278 (1990) (Also in Collected Papers, vol. 2, pp. 15–45. Springer, Springer, Berlin/Heidelberg (1991))Google Scholar
  30. 124.
    Strömberg, F.: Computational aspects of Maass waveforms. Ph.D. thesis, Uppsala University (2004)Google Scholar
  31. 125.
    Strömberg, F.: Computation of Maass waveforms with non-trivial multiplier systems. Syst. Math. Comput. 77, 2375–2416 (2008)CrossRefzbMATHGoogle Scholar
  32. 126.
    Strömberg, F.: Computation of Selberg Zeta Functions on Hecke Triangle Groups. arXiv:0804.4837v1 (2008, preprint)Google Scholar
  33. 127.
    Strömberg, F.: Maass Waveforms on (Γ 0(N), χ) (Computational Aspects). In: Bolte, J., Steiner, F. (eds.) Hyperbolic Geometry and Applications in Quantum Chaos and Cosmology, pp. 187–228. Cambridge University Press, Cambridge, UK (2011)Google Scholar
  34. 129.
    Then, H.: Maass cusp forms for large eigenvalues. Math. Comput. 74 (249), 363–381 (2005)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Markus Szymon Fraczek
    • 1
  1. 1.Mathematics InstituteUniversity of WarwickCoventryUK

Personalised recommendations