Asymptotics for the parabolic, hyperbolic, and elliptic Eisenstein series through hyperbolic and elliptic degeneration
Item
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Title
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Asymptotics for the parabolic, hyperbolic, and elliptic Eisenstein series through hyperbolic and elliptic degeneration
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Identifier
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d_2009_2013:5f55b3c8e5e7:10099
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identifier
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10269
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Creator
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Garbin, Daniel J.,
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Contributor
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Jay Jorgenson
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Date
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2009
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Language
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English
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Publisher
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City University of New York.
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Subject
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Mathematics | degeneration | Eisenstein series | Riemann surfaces
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Abstract
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Let Gamma be a Fuchsian group of the first kind acting on the hyperbolic upper half plane H , and let M = Gamma\ H be the associated (connected) finite volume hyperbolic Riemann surface. We will allow the presence of both parabolic and elliptic elements as part of the group Gamma, so that the surface has cusps (coming from parabolic elements) and conical points (coming from elliptic elements). To each primitive Gamma-inconjugate parabolic element there is an associated parabolic Eisentein series which is more commonly referred to in the literature as the non-holomorphic Eisenstein series. If gamma is a primitive Gamma-inconjugate hyperbolic element, then following the work due to Kudla and Millson, there is an associated hyperbolic Eisenstein series. More recently, Jorgenson and Kramer have introduced an elliptic Eisenstein series associated to a primitive Gamma-inconjugate elliptic element gamma of the discontinous group Gamma.;In this note, we look at the behavior of these Eisenstein series on families of hyperbolic Riemann surfaces of finite volume. In particular, there are two types of families that we study. The first family is obtained by hyperbolic degeneration which is a process that involves pinching primitive simple closed geodesic. The second family is obtained by elliptic degeneration, a process in which the order of ramification becomes unbounded, namely the order of elliptic fixed points associated to the conical points of the surface runs off to infinity. The main results are as follows. The Eisenstein series that are not associated to degenerating elements will converge to their correspondents in the limiting surface. For the Eisenstein series that are associated to degenerating elements the situation is as follows. In the case of hyperbolic degeneration, the hyperbolic Eisenstein series associated to a pinched geodesic will converge (up to a multiplicative factor) to the parabolic Eisenstein series associated to the newly developed cusp(s) in the limit surface. In the case of elliptic degeneration, a strikingly similar result occurs since the elliptic Eisenstein series associated to a degenerating conical point converges (up to a multiplicative factor) to the parabolic Eisenstein series associated to the newly developed cusp in the limit surface. The striking similarity lays in the fact that the above multiplicative factors involve the parameters defining the two type of degeneration, namely the length of the pinched geodesic in the case of hyperbolic degeneration and the angle of the pinched cone in the case of elliptic degeneration.
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Type
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dissertation
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Source
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2009_2013.csv
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degree
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Ph.D.
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Program
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Mathematics
