On the appearance of Eisenstein series through degeneration

• Autores: Daniel Garbin, Jay Jorgenson, Michael Munn
• Localización: Commentarii mathematici helvetici, ISSN 0010-2571, Vol. 83, Nº 4, 2008, págs. 701-721
• Idioma: inglés
• DOI: 10.4171/cmh/140
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• Resumen

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