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

DOI:
https://doi.org/10.1090/S0002-9947-02-03109-4

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 -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 -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.

**[1]**Efrat, Isaac Y.,*The Selberg trace formula for*, Mem. Amer. Math. Soc. 65 (1987), no. 359, iv+111 pp. MR**88e:11041****[2]**Efrat, I. and Sarnak, P.,*The determinant of the Eisenstein matrix and Hilbert class fields*, Trans. Amer. Math. Soc. 290 (1985), no. 2, 815-824. MR**87b:11039****[3]**Janusz, Gerald J.,*Algebraic number fields*, Pure and Applied Mathematics, Vol. 55. Academic Press [A Subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1973. x+220 pp. MR**51:3110****[4]**Kubota, Tomio,*Elementary theory of Eisenstein series*, Kodansha Ltd., Tokyo; Halsted Press [John Wiley and Sons], New York-London-Sydney, 1973. xi+110 pp. MR**55:2759****[5]**Neukirch, Jürgen,*Algebraic number theory*, Translated from the 1992 German original and with a note by Norbert Schappacher. With a foreword by G. Harder. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 322. Springer-Verlag, Berlin, 1999. xviii+571 pp. MR**2000m:11104****[6]**Selberg, A.,*Harmonic analysis and discontinuous groups in weakly symmetric Riemannian spaces with applications to Dirichlet series*, J. Indian Math. Soc. (N.S.) 20 (1956), 47-87. MR**19:531g****[7]**Selberg, Atle,*Recent developments in the theory of discontinuous groups of motions of symmetric spaces*, 1970 Proceedings of the Fifteenth Scandinavian Congress (Oslo, 1968), Lecture Notes in Mathematics, Vol. 118 Springer, Berlin, pp. 99-120 MR**41:8595**

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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

Email:
mazanti@imf.aau.dk

DOI:
https://doi.org/10.1090/S0002-9947-02-03109-4

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