Skip to main content
SpringerLink
Log in

The Hyperbolic Laplace-Beltrami Operator

  • Chapter
  • First Online:
Selberg Zeta Functions and Transfer Operators

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2139))

Abstract

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].

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 54.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 69.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Abramowitz, M., Stegun, I.: Handbook of Mathematical Functions. Dover, New York (1964)

    MATH  Google Scholar 

  2. Avelin, H.: Deformation of Γ 0(5)-cusp forms. Math. Comput. 76, 361–384 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Avelin, H.: Computations of Eisenstein series on Fuchsian groups. Math. Comput. 77, 1779–1800 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  4. Booker, A., Strömbergsson, A., Venkatesh, A.: Effective computation of Maass cusp forms. IMRN 2006, 1–34 (2006)

    MATH  Google Scholar 

  5. Borthwick, D.: Introduction to spectral theory on hyperbolic surfaces. Proc. Symp. Pure Math. 84, 3–48 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bruggeman, R.: Families of Automorphic Forms. Birkhäuser, Basel/Boston (1994)

    Book  MATH  Google Scholar 

  7. 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)

    MathSciNet  MATH  Google Scholar 

  8. Bruggeman, R., Lewis, J., Zagier, D.: Period Functions for Maass Forms and Cohomology. Memoirs of the American Mathematical Society. American Mathematical Society, Providence (2015)

    MATH  Google Scholar 

  9. Bump, D.: Automorphic Forms and Representations. Cambridge University Press, Cambridge/New York (1998)

    MATH  Google Scholar 

  10. 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. Deitmar, A., Hilgert, J.: A Lewis correspondence for submodular groups. Forum Mathematicum 19 (6), 1075–1099 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  12. Fraczek, M.: Spezielle Eigenfunktionen des Transfer-Operators für Hecke Kongruenz Untergruppen. Diploma Thesis, Clausthal University (2006)

    Google Scholar 

  13. 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)

    Article  MathSciNet  MATH  Google Scholar 

  14. 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)

    MathSciNet  MATH  Google Scholar 

  15. Guillope, L., Lin, K., Zworski, M.: The Selberg zeta function for convex co-compact Schottky groups. Commun. Math. Phys. 245 (1), 149–175 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  16. 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. 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. 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)

    Chapter  Google Scholar 

  19. 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)

    Article  MathSciNet  MATH  Google Scholar 

  20. 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. Iwaniec, H.: Spectral Methods of Automorphic Forms. American Mathematical Society, Providence (2002)

    Book  MATH  Google Scholar 

  22. Lewis, J., Zagier, D.: Period functions for Maass wave forms, I. Ann. Math. 153, 191–258 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  23. 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)

    Chapter  Google Scholar 

  24. Matthies, C., Steiner, F.: Selberg’s ζ function and the quantization of chaos. Phys. Rev. A 44 (12), R7877–R7880 (1991)

    Article  MathSciNet  Google Scholar 

  25. Miyake, T.: Modular Forms. Springer Monographs in Mathematics. Springer, Berlin/New York (2006)

    MATH  Google Scholar 

  26. Mühlenbruch, T.: Hecke operators on period functions for Γ 0(n). J. Number Theory 118, 208–235 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  27. Phillips, R., Sarnak, P.: The spectrum of fermat curves. Geom. Funct. Anal. 1, 80–146 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  28. Phillips, R., Sarnak, P.: Cusp forms for character varieties. Geom. Funct. Anal. 4, 93–118 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  29. 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. Strömberg, F.: Computational aspects of Maass waveforms. Ph.D. thesis, Uppsala University (2004)

    Google Scholar 

  31. Strömberg, F.: Computation of Maass waveforms with non-trivial multiplier systems. Syst. Math. Comput. 77, 2375–2416 (2008)

    Article  MATH  Google Scholar 

  32. Strömberg, F.: Computation of Selberg Zeta Functions on Hecke Triangle Groups. arXiv:0804.4837v1 (2008, preprint)

    Google Scholar 

  33. 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. Then, H.: Maass cusp forms for large eigenvalues. Math. Comput. 74 (249), 363–381 (2005)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematics Institute, University of Warwick, Coventry, UK

    Markus Szymon Fraczek

Authors
  1. Markus Szymon Fraczek
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this chapter

Cite this chapter

Fraczek, M.S. (2017). The Hyperbolic Laplace-Beltrami Operator. In: Selberg Zeta Functions and Transfer Operators. Lecture Notes in Mathematics, vol 2139. Springer, Cham. https://doi.org/10.1007/978-3-319-51296-9_6

Download citation

Publish with us

Policies and ethics

Search

Navigation