JournalscmhVol. 83, No. 4pp. 701–721

On the appearance of Eisenstein series through degeneration

  • Daniel Garbin

    The Graduate Center of CUNY, New York, United States
  • Jay Jorgenson

    The City College of New York - CUNY, USA
  • Michael Munn

    The Graduate Center of CUNY, New York, United States
On the appearance of Eisenstein series through degeneration cover
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Abstract

Let Γ be a Fuchsian group of the first kind acting on the hyperbolic upper half plane ℍ, and let M = Γ \ ℍ be the associated finite volume hyperbolic Riemann surface. If γ is parabolic, there is an associated (parabolic) Eisenstein series, which, by now, is a classical part of mathematical literature. If γ 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 γ ∈ Γ corresponds to a degenerating hyperbolic element, then a multiple of the associated hyperbolic Eisenstein series converges to parabolic Eisenstein series on the limit surface.

Cite this article

Daniel Garbin, Jay Jorgenson, Michael Munn, On the appearance of Eisenstein series through degeneration. Comment. Math. Helv. 83 (2008), no. 4, pp. 701–721

DOI 10.4171/CMH/140