Mathematische Annalen

, Volume 346, Issue 4, pp 931–947 | Cite as

On the spectral expansion of hyperbolic Eisenstein series

  • Jay Jorgenson
  • Jürg KramerEmail author
  • Anna-Maria von Pippich


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.


Riemann Surface Eisenstein Series Fuchsian Group Geodesic Path Laurent Expansion 
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Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  • Jay Jorgenson
    • 1
  • Jürg Kramer
    • 2
    Email author
  • Anna-Maria von Pippich
    • 2
  1. 1.Department of MathematicsThe City College of New YorkNew YorkUSA
  2. 2.Institut für MathematikHumboldt-Universität zu BerlinBerlinGermany

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