Skip to main content
SpringerLink
Log in

Introduction

  • Chapter
  • First Online:
The Spectrum of Hyperbolic Surfaces

Part of the book series: Universitext ((UTX))

  • 2560 Accesses

Abstract

Before entering into the heart of the subject, we begin with the motivating and familiar example of spectral analysis on the torus.

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 59.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 79.99
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

Notes

  1. 1.

    We consider the negative of the usual Laplacian in order for its spectrum to lie in R +. One sometimes calls this operator the “geometric Laplacian”.

  2. 2.

    Throughout this book we shall use the notation e(z) = e 2π i z.

  3. 3.

    Let (M, g) be a Riemannian manifold and x a point in M. Given two tangent vectors v, w ∈ T x (M) of norm 1, we have \(g_{x}(v,w) =\cos \theta\) where \(\theta\) is the angle between v and w.

  4. 4.

    We in fact made indirect use of the unit disk model to show that the hyperbolic circles are Euclidean circles.

  5. 5.

    Figure 1.3 was realized by McMullen, see http://www.math.harvard.edu/~ctm/gallery/index.html.

  6. 6.

    There are quite a few works dedicated to the holomorphic theory [70, 92, 118]. Although many ideas in this text were originally developed in the context of holomorphic modular forms and are simply extended here to Maaß forms, we wanted to focus our attention on analytic questions and have therefore not recalled – even briefly – this beautiful theory.

  7. 7.

    The norm of a closed geodesic is the exponential of its length. The prime closed geodesics are the closed geodesics which are traced out just once.

  8. 8.

    We sketch a proof of the prime number theorem in Chap. 4. We refer the reader to the book of Titchmarsh [129] for a complete proof of this theorem, its link with the Riemann zeta function and the Riemann hypothesis.

References

  1. Alan F. Beardon, The geometry of discrete groups, Graduate Texts in Mathematics, vol. 91, Springer-Verlag, New York, 1995, Corrected reprint of the 1983 original.

    Google Scholar 

  2. Jens Bolte and Stefan Johansson, A spectral correspondence for Maaß waveforms, Geom. Funct. Anal. 9 (1999), no. 6, 1128–1155.

    Article  MathSciNet  MATH  Google Scholar 

  3. Daniel Bump, W., Geometry and spectra of compact Riemann surfaces, Progress in Mathematics, vol. 106, Birkhäuser Boston Inc., Boston, MA, 1992.

    Google Scholar 

  4. Henri Paul de Saint-Gervais, Uniformisation des surfaces de Riemann. Retour sur un théorème centenaire, ENS Éditions, Lyon, 2010.

    Google Scholar 

  5. Régine et Adrien Douady, Algèbre et théories galoisiennes, Cassini, 2005.

    Google Scholar 

  6. Lester R. Ford, Automorphic functions, Chelsea, 2004.

    Google Scholar 

  7. Sylvestre Gallot, Dominique Hulin, and Jacques Lafontaine, Riemannian geometry, third ed., Universitext, Springer-Verlag, Berlin, 2004.

    Book  MATH  Google Scholar 

  8. Alex Gamburd, On the spectral gap for infinite index “congruence” subgroups of SL2(Z), Israel J. Math. 127 (2002), 157–200.

    Article  MathSciNet  Google Scholar 

  9. Stephen S. Gelbart, Automorphic forms on adèle groups, Princeton University Press, Princeton, N.J., 1975, Annals of Mathematics Studies, No. 83.

    Google Scholar 

  10. Étienne Ghys, Poincaré and his disk, The scientific legacy of Poincaré (Éric Charpentier, Étienne Ghys, and Annick Lesne, eds.), Hist. Math., vol. 36, American Mathematical Society, Providence, RI, 2010, Translated from the French original (2006), pp. 17–46.

    Google Scholar 

  11. Dennis A. Hejhal, The Selberg trace formula and the Riemann zeta function, Duke Math. J. 43 (1976), no. 3, 441–482.

    Article  MathSciNet  MATH  Google Scholar 

  12. J.-M. Deshouillers and, The Selberg trace formula for PSL (2, R ). Vol. I, Lecture Notes in Mathematics, vol. 548, Springer-Verlag, Berlin, 1976.

    Google Scholar 

  13. Sigurdur Helgason, Topics in harmonic analysis on homogeneous spaces, Progress in Mathematics, vol. 13, Birkhäuser Boston, Mass., 1981.

    Google Scholar 

  14. Heinz Huber, Zur analytischen Theorie hyperbolischer Raumformen und Bewegungsgruppen. II, Math. Ann. 142 (1960/1961), 385–398.

    Google Scholar 

  15. Henryk Iwaniec, Prime geodesic theorem, J. Reine Angew. Math. 349 (1984), 136–159.

    MathSciNet  MATH  Google Scholar 

  16. M. N. Huxley, Spectral methods of automorphic forms, second ed., Graduate Studies in Mathematics, vol. 53, American Mathematical Society, Providence, RI, 2002.

    Google Scholar 

  17. H. Jacquet and R. P. Langlands, Automorphic forms on GL (2), Lecture Notes in Mathematics, vol. 114, Springer-Verlag, Berlin, 1970.

    Google Scholar 

  18. Svetlana Katok, Fuchsian groups, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1992.

    Google Scholar 

  19. Henry H. Kim, Functoriality for the exterior square of GL4 and the symmetric fourth of GL2, J. Amer. Math. Soc. 16 (2003), no. 1, 139–183 (electronic), With Appendix 1 by Dinakar Ramakrishnan and Appendix 2 by Kim and Peter Sarnak.

    Google Scholar 

  20. Neal Koblitz, Introduction to elliptic curves and modular forms, second ed., Graduate Texts in Mathematics, vol. 97, Springer-Verlag, New York, 1993.

    Google Scholar 

  21. N. V. Kuznetsov, Arifmeticheskaya forma formuly sleda Selberga i raspredelenie norm primitivnykh giperbolicheskikh klassov modulyarnoi gruppy, Akad. Nauk SSSR Dal′nevostoč. Naučn. Centr, 1978, Presented at the Far Eastern Mathematical School.

    Google Scholar 

  22. J. Lewis and D. Zagier, Period functions for Maass wave forms. I, Ann. of Math. (2) 153 (2001), no. 1, 191–258.

    Google Scholar 

  23. Elon Lindenstrauss and Akshay Venkatesh, Existence and Weyl’s law for spherical cusp forms, Geom. Funct. Anal. 17 (2007), no. 1, 220–251.

    Article  MathSciNet  MATH  Google Scholar 

  24. W. Luo, Z. Rudnick, and P. Sarnak, On Selberg’s eigenvalue conjecture, Geom. Funct. Anal. 5 (1995), no. 2, 387–401.

    Article  MathSciNet  MATH  Google Scholar 

  25. William S. Massey, A basic course in algebraic topology, Graduate Texts in Mathematics, vol. 127, Springer-Verlag, New York, 1991.

    Google Scholar 

  26. Toshitsune Miyake, Modular forms, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2006, Translated from the 1976 Japanese original by Yoshitaka Maeda.

    Google Scholar 

  27. Georg Friedrich Bernhard Riemann, Œuvres mathématiques de Riemann, Traduites de l’allemand par L. Laugel, avec une préface de C. Hermite et un discours de Félix Klein. Nouveau tirage, Librairie Scientifique et Technique Albert Blanchard, Paris, 1968.

    Google Scholar 

  28. Walter Roelcke, Über die Wellengleichung bei Grenzkreisgruppen erster Art, S.-B. Heidelberger Akad. Wiss. Math.-Nat. Kl. 1953/1955 (1953/1955), 159–267 (1956).

    Google Scholar 

  29. Walter Rudin, Real and complex analysis, third ed., McGraw-Hill Book Co., New York, 1987.

    MATH  Google Scholar 

  30. Frigyes Riesz and Béla Sz.-, Spectra of hyperbolic surfaces, Bull. Amer. Math. Soc. (N.S.) 40 (2003), no. 4, 441–478 (electronic).

    Google Scholar 

  31. Peter Sarnak and Xiao Xi Xue, Bounds for multiplicities of automorphic representations, Duke Math. J. 64 (1991), no. 1, 207–227.

    Google Scholar 

  32. Frigyes Riesz and Béla Sz.-, On the estimation of Fourier coefficients of modular forms, Proc. Sympos. Pure Math., Vol. VIII, American Mathematical Society, Providence, R.I., 1965, pp. 1–15.

    Google Scholar 

  33. Frigyes Riesz and Béla Sz.-, Cours d’arithmétique, Le Mathématicien, vol. 2, Presses Universitaires de France, Paris, 1977, Deuxième édition revue et corrigée.

    Google Scholar 

  34. Frigyes Riesz and Béla Sz.-, Advanced analytic number theory, second ed., Tata Institute of Fundamental Research Studies in Mathematics, vol. 9, Tata Institute of Fundamental Research, Bombay, 1980.

    Google Scholar 

  35. E. C. Titchmarsh, The theory of the Riemann zeta-function, second ed., The Clarendon Press Oxford University Press, New York, 1986.

    MATH  Google Scholar 

  36. A. B. Venkov, Spectral theory of automorphic functions, Trudy Mat. Inst. Steklov. 153 (1981), 172.

    MathSciNet  Google Scholar 

  37. A. B. Venkov, Quelques remarques sur la conjecture λ 1 ≥ 1∕4, Seminar on number theory, Paris 1981–82 (Paris, 1981/1982), Progr. Math., vol. 38, Birkhäuser Boston, Boston, MA, 1983, pp. 321–343.

    Google Scholar 

  38. Claude Zuily and Hervé Queffélec, Analyse pour l’agrégation, third ed., Sciences Sup, Dunod, Paris, 2007.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. IMJ-PRG, Universite Pierre et Marie Curie, Paris, France

    Nicolas Bergeron

Authors
  1. Nicolas Bergeron
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Bergeron, N. (2016). Introduction. In: The Spectrum of Hyperbolic Surfaces. Universitext. Springer, Cham. https://doi.org/10.1007/978-3-319-27666-3_1

Download citation

Publish with us

Policies and ethics

Search

Navigation