On the spectral expansion of hyperbolic Eisenstein series |
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Authors: | Jay Jorgenson Jürg Kramer Anna-Maria von Pippich |
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Institution: | 1. Department of Mathematics, The City College of New York, Convent Ave. at 138th Street, New York, NY, 10031, USA 2. Institut für Mathematik, Humboldt-Universit?t zu Berlin, Unter den Linden 6, 10099, Berlin, Germany
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Abstract: | In this article, we determine the spectral expansion, meromorphic continuation, and location of poles with identifiable singularities
for the scalar-valued hyperbolic Eisenstein series. Similar to the form-valued hyperbolic Eisenstein series studied in Kudla
and Millson (Invent Math 54:193–211, 1979), the scalar-valued hyperbolic Eisenstein series is defined for each primitive,
hyperbolic conjugacy class within the uniformizing group associated to any finite volume hyperbolic Riemann surface. Going
beyond the results in Kudla and Millson (Invent Math 54:193–211, 1979) and Risager (Int Math Res Not 41:2125–2146, 2004),
we establish a precise spectral expansion for the hyperbolic Eisenstein series for any finite volume hyperbolic Riemann surface
by first proving that the hyperbolic Eisenstein series is in L
2. Our other results, such as meromorphic continuation and determination of singularities, are derived from the spectral expansion. |
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Keywords: | |
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