Skip to main content
SpringerLink
Log in

Dégénérescence de séries d’Eisenstein hyperboliques

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

In this work, we deal with hyperbolic Eisenstein series and more particularly, given a degenerating family S l (l ≥ 0) of Riemann Surfaces with their canonical hyperbolic metrics, we work out the degeneration of hyperbolic Eisenstein series associated to the pinching geodesics in S l . Our principal results are theorems 2.2, 4.1 et 4.2.

Résumé

Dans ce travail, nous nous intéressons aux séries d’Eisenstein hyperboliques et plus particulièrement à leur dégénérescence. On considère une famille de surfaces de Riemann hyperboliques compactes dégénérant, en pinçant une géodésique, vers une surface de Riemann à pointes S 0. Nos résultats principaux sont les Théorèmes 2.2, 4.1 et 4.2.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Bers L. (1974). Spaces of degenerating Riemann surfaces. Ann. Math. Stud. 79: 43–55

    Google Scholar 

  2. Buser, P.: Geometry and spectra of compact Riemann surfaces, vol. 106. Progress in Mathematics. Birkhauser, Boston (1992)

    Google Scholar 

  3. Colbois B. and Courtois G. (1989). Les valeurs propres inférieures à 1/4 des surfaces de Riemann de petit rayon d’injectivité. Comment. Math. Helv. 64: 349–362

    Article  MATH  Google Scholar 

  4. Falliero Th. (2000). Décomposition spectrale de 1-formes différentielles sur une surface de Riemann et séries d’Eisenstein. Math. Ann. 317: 263–284

    Article  MATH  Google Scholar 

  5. Fay, J.D.: Theta functions on Riemann surfaces, vol. 352. Lecture Notes in Mathematics. Springer, Heidelberg (1973)

    Google Scholar 

  6. Fay J.D. (1977). Fourier coefficients of the resolvent for a Fuchsian group. J. Reine Angew. Math. 293: 143–203

    Google Scholar 

  7. Gérardin, P.: Formes automorphes associées aux cycles géodésiques des surfaces de Riemann hyperboliques. Lecture Notes in Mathematics, vol. 901, Novembre (1980)

  8. Hejhal, D.A.: The Selberg Trace Formula for PSL(2,R), vol. 1. Springer Lecture Notes 548 (1976)

  9. Hejhal, D.A.: The Selberg Trace Formula for PSL(2,R), vol. 2. Springer Lecture Notes 1001 (1983)

  10. Hejhal D.A. (1984). Modular Forms A continuity method for spectral theory on Fuchsian groups. Ellis-Horwood, Chichester

    Google Scholar 

  11. Hejhal, D.A.: Regular b-groups, degenerating Riemann surfaces and spectral theory. Mem. Am. Math. Soc. 88(437) (1990)

  12. Imayoshi Y. and Taniguchi M. (1992). An Introduction to Teichmüller Spaces. Springer, Heidelberg

    MATH  Google Scholar 

  13. Iwaniec, H.: Introduction to the Spectral Theory of Automorphic Forms. Biblioteca de la revista matematica iberoamericana (1995)

  14. Ji L. (1993). The asymptotic behavior of Green’s functions for degenerating hyperbolic surfaces. Math. Z. 212: 375–394

    Article  MATH  Google Scholar 

  15. Ji L. (1993). Spectral degeneration of hyperbolic Riemann surfaces. J. Differ. Geom. 38: 263–313

    MATH  Google Scholar 

  16. Keen L. (1973). Collars on Riemann surfaces. Ann. Math. Stud. 79: 263–268

    Google Scholar 

  17. Kubota T. (1973). Elementary Theory of Eisenstein Series. Halsted Press, New York

    MATH  Google Scholar 

  18. Kudla J. and Millson S. (1979). Harmonic differentials and closed geodesics on a Riemann surface. Invent. Math. 54: 193–211

    Article  MATH  Google Scholar 

  19. Lebowitz A. (1971). On the degeneration of Riemann surfaces. Advances in the theory of Riemann surfaces. Ann. Math. Stud. 66: 265–286

    Google Scholar 

  20. Petersson H. (1943). Ein Summationsverfahren für die Poincaréschen Reihen von der Dimension-2 zu den hyperbolischen Fixpunktepaaren. Math. Zeit. 49: 441–446

    Article  Google Scholar 

  21. Randol B. (1979). Cylinders in riemannian surfaces. Comment. Math. Helv. 54: 1–5

    Article  MATH  Google Scholar 

  22. Schiffer M. and Spencer D.C. (1954). Functionals of finite Riemann surfaces. Princeton University Press, Princeton

    MATH  Google Scholar 

  23. Wolpert S.A. (1987). Asymptotics of the spectrum and the Selberg Zeta function on the space of Riemann surfaces. Commun. Math. Phys. 112: 283–315

    Article  MATH  Google Scholar 

  24. Wolpert S.A. (1992). Spectral limits for hyperbolic surfaces, II. Invent. Math. 108: 91–129

    Article  MATH  Google Scholar 

  25. Yamada A. (1980). Precise variational formulas for abelian differentials. Kodai Math. J. 3: 114–143

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculté des Sciences, Université d’Avignon, 33 rue Louis Pasteur, 84000, Avignon, France

    Thérèse Falliero

Authors
  1. Thérèse Falliero
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Thérèse Falliero.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Falliero, T. Dégénérescence de séries d’Eisenstein hyperboliques. Math. Ann. 339, 341–375 (2007). https://doi.org/10.1007/s00208-007-0116-0

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-007-0116-0

Mathematics Subject Classification (2000)

Search

Navigation