On the Behavior of Eisenstein Series Through Elliptic Degeneration

Let Gamma be a Fuchsian group of the first kind acting on the hyperbolic upper half plane H, and let M = Gamma backslash H be the associated finite volume hyperbolic Riemann surface. If gamma is a primitive parabolic, hyperbolic, resp. elliptic element of Gamma, 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.