On the appearance of Eisenstein series through degeneration

Abstract

Let $\Gamma$ 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 $\gamma$ is parabolic, there is an associated (parabolic) Eisenstein series, which, by now, is a classical part of mathematical literature. If $\gamma$ is hyperbolic, then, following ideas due to Kudla-Millson, there is a corresponding hyperbolic Eisenstein series. In this article, we study the limiting behavior of parabolic and hyperbolic Eisenstein series on a degenerating family of finite volume hyperbolic Riemann surfaces. In particular, we prove the following result. If $\gamma \in \Gamma$ corresponds to a degenerating hyperbolic element, then a multiple of the associated hyperbolic Eisenstein series converges to parabolic Eisenstein series on the limit surface.

15 pages, 2 figures. This paper has been accepted for publication in Commentarii Mathematici Helvetici

References in corpus (3)

• Parabolic, hyperbolic and elliptic Poincaré series
• Distribution of modular symbols for compact surfaces
• Asymptotics of degenerating Eisenstein series

Cited by in corpus (2)

• A Kronecker limit type formula for elliptic Eisenstein series
• On the minimal distance between elliptic fixed points for geometrically-finite Fuchsian groups