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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
Article

Abstract

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.

Keywords

Riemann Surface Homotopy Class Fundamental Domain Eisenstein Series Counting Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© 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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