Skip to main content
SpringerLink
Log in

On the spectral expansion of hyperbolic Eisenstein series

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

In this article, we determine the spectral expansion, meromorphic continuation, and location of poles with identifiable singularities for the scalar-valued hyperbolic Eisenstein series. Similar to the form-valued hyperbolic Eisenstein series studied in Kudla and Millson (Invent Math 54:193–211, 1979), the scalar-valued hyperbolic Eisenstein series is defined for each primitive, hyperbolic conjugacy class within the uniformizing group associated to any finite volume hyperbolic Riemann surface. Going beyond the results in Kudla and Millson (Invent Math 54:193–211, 1979) and Risager (Int Math Res Not 41:2125–2146, 2004), we establish a precise spectral expansion for the hyperbolic Eisenstein series for any finite volume hyperbolic Riemann surface by first proving that the hyperbolic Eisenstein series is in L 2. Our other results, such as meromorphic continuation and determination of singularities, are derived from the spectral expansion.

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

Similar content being viewed by others

References

  1. Chavel I.: Eigenvalues in Riemannian Geometry. Academic Press, New York (1984)

    MATH  Google Scholar 

  2. Falliero T.: Dégénérescence de séries d’Eisenstein hyperboliques. Math. Ann. 339, 341–375 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  3. Garbin D., Jorgenson J., Munn M.: On the appearance of Eisenstein series through degeneration. Comment. Math. Helv. 83, 701–721 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  4. Iwaniec, H.: Spectral methods of automorphic forms. Graduate Studies in Mathematics, vol. 53. Amer. Math. Soc., Providence (2002)

  5. Jorgenson J., Lundelius R.: Convergence of the normalized spectral counting function on degenerating hyperbolic Riemann surfaces of finite volume. J. Funct. Anal. 149, 25–57 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  6. Jorgenson J., O’Sullivan C.: Unipotent vector bundles and higher-order non-holomorphic Eisenstein series. J. Théor. Nombres Bordeaux 20, 131–163 (2008)

    MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  8. Lundelius R.: Asymptotics of the determinant of the Laplacian on hyperbolic surfaces of finite volume. Duke Math. J. 71, 212–242 (1993)

    Article  MathSciNet  Google Scholar 

  9. O’Sullivan C.: Properties of Eisenstein series formed with modular symbols. J. Reine Angew. Math. 518, 163–186 (2000)

    MATH  MathSciNet  Google Scholar 

  10. Risager M.: On the distribution of modular symbols for compact surfaces. Int. Math. Res. Not. 41, 2125–2146 (2004)

    Article  MathSciNet  Google Scholar 

  11. Pippich A.-M.: Elliptische Eisensteinreihen. Diplomarbeit, Humboldt-Universität zu Berlin (2005)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, The City College of New York, Convent Ave. at 138th Street, New York, NY, 10031, USA

    Jay Jorgenson

  2. Institut für Mathematik, Humboldt-Universität zu Berlin, Unter den Linden 6, 10099, Berlin, Germany

    Jürg Kramer & Anna-Maria von Pippich

Authors
  1. Jay Jorgenson
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jürg Kramer
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Anna-Maria von Pippich
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jürg Kramer.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Jorgenson, J., Kramer, J. & von Pippich, AM. On the spectral expansion of hyperbolic Eisenstein series. Math. Ann. 346, 931–947 (2010). https://doi.org/10.1007/s00208-009-0422-9

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-009-0422-9

Keywords

Search

Navigation