On the appearance of Eisenstein series through degeneration

TL;DR

This paper investigates how hyperbolic and parabolic Eisenstein series behave and transform during the degeneration of hyperbolic Riemann surfaces, revealing that hyperbolic series converge to parabolic series in the limit.

Contribution

It establishes the convergence of hyperbolic Eisenstein series to parabolic Eisenstein series on degenerating hyperbolic surfaces, connecting two classical types of Eisenstein series.

Findings

Hyperbolic Eisenstein series converge to parabolic Eisenstein series during degeneration.

The study provides a precise limiting behavior of Eisenstein series on degenerating surfaces.

Results extend understanding of Eisenstein series in degenerating geometric contexts.

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.