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Fourier expansion of Eisenstein series on the Hilbert modular group and Hilbert class fields

Author: Claus Mazanti Sorensen
Journal: Trans. Amer. Math. Soc. 354 (2002), 4847-4869
MSC (2000): Primary 11F30, 11F41, 11M36, 11R37, 11R42
Published electronically: August 1, 2002
MathSciNet review: 1926839
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Abstract: In this paper we consider the Eisenstein series for the Hilbert modular group of a general number field. We compute the Fourier expansion at each cusp explicitly. The Fourier coefficients are given in terms of completed partial Hecke $L$-series, and from their functional equations, we get the functional equation for the Eisenstein vector. That is, we identify the scattering matrix. When we compute the determinant of the scattering matrix in the principal case, the Dedekind $\xi$-function of the Hilbert class field shows up. A proof in the imaginary quadratic case was given in Efrat and Sarnak, and for totally real fields with class number one a proof was given in Efrat.

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Additional Information

Claus Mazanti Sorensen
Affiliation: Department of Mathematics, Ny Munkegade, 8000 Aarhus C, Denmark
Address at time of publication: Department of Mathematics, California Institute of Technology, Pasadena, California 91125, USA

Keywords: Eisenstein series, Hilbert modular groups, Hilbert class fields
Received by editor(s): March 26, 2002
Received by editor(s) in revised form: May 13, 2002
Published electronically: August 1, 2002
Article copyright: © Copyright 2002 American Mathematical Society

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