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), 48474869
MSC (2000):
Primary 11F30, 11F41, 11M36, 11R37, 11R42
Published electronically:
August 1, 2002
MathSciNet review:
1926839
Fulltext PDF Free Access
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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]
Isaac
Y. Efrat, The Selberg trace formula for
𝑃𝑆𝐿₂(𝑅)ⁿ, Mem. Amer.
Math. Soc. 65 (1987), no. 359, iv+111. MR 874084
(88e:11041), http://dx.doi.org/10.1090/memo/0359
 [2]
I.
Efrat and P.
Sarnak, The determinant of the Eisenstein
matrix and Hilbert class fields, Trans. Amer.
Math. Soc. 290 (1985), no. 2, 815–824. MR 792829
(87b:11039), http://dx.doi.org/10.1090/S00029947198507928291
 [3]
Gerald
J. Janusz, Algebraic number fields, Academic Press [A
Subsidiary of Harcourt Brace Jovanovich, Publishers], New YorkLondon,
1973. Pure and Applied Mathematics, Vol. 55. MR 0366864
(51 #3110)
 [4]
Tomio
Kubota, Elementary theory of Eisenstein series, Kodansha Ltd.,
Tokyo; Halsted Press [John Wiley & Sons], New YorkLondonSydney, 1973.
MR
0429749 (55 #2759)
 [5]
Jürgen
Neukirch, Algebraic number theory, Grundlehren der
Mathematischen Wissenschaften [Fundamental Principles of Mathematical
Sciences], vol. 322, SpringerVerlag, Berlin, 1999. Translated from
the 1992 German original and with a note by Norbert Schappacher; With a
foreword by G. Harder. MR 1697859
(2000m:11104)
 [6]
A.
Selberg, 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 0088511
(19,531g)
 [7]
Atle
Selberg, Recent developments in the theory of discontinuous groups
of motions of symmetric spaces, Proceedings of the Fifteenth
Scandinavian Congress (Oslo, 1968) Springer, Berlin, 1970,
pp. 99–120. MR 0263996
(41 #8595)
 [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, 815824. 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 YorkLondon, 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 YorkLondonSydney, 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. SpringerVerlag, 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), 4787. 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. 99120 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:
http://dx.doi.org/10.1090/S0002994702031094
PII:
S 00029947(02)031094
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
