Communications in Mathematical Physics

, Volume 292, Issue 2, pp 511–528 | Cite as

On the Behavior of Eisenstein Series Through Elliptic Degeneration

  • D. GarbinEmail author
  • A.-M. v. Pippich


Let Γ be a Fuchsian group of the first kind acting on the hyperbolic upper half plane \({\mathbb{H}}\), and let \({M = \Gamma\backslash \mathbb{H}}\) be the associated finite volume hyperbolic Riemann surface. If γ is a primitive parabolic, hyperbolic, resp. elliptic element of Γ, there is an associated parabolic, hyperbolic, resp. elliptic Eisenstein series. In this article, we study the limiting behavior of these Eisenstein series on an elliptically degenerating family of finite volume hyperbolic Riemann surfaces. In particular, we prove the following result. The elliptic Eisenstein series associated to a degenerating elliptic element converges up to a factor to the parabolic Eisenstein series associated to the parabolic element which fixes the newly developed cusp on the limit surface.


Riemann Surface Homotopy Class Fundamental Domain Eisenstein Series Counting Function 
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© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Mathematics Ph.D. ProgramThe Graduate Center of CUNYNew YorkU.S.A.
  2. 2.Mathematics Ph.D. ProgramHumboldt-Universität zu Berlin, Institut für MathematikBerlinGermany

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