The scattering matrix for the Hilbert modular group

Masri, Riad

doi:10.1090/S0002-9939-09-09800-1

The scattering matrix for the Hilbert modular group
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Proc. Amer. Math. Soc. 137 (2009), 2541-2555 Request permission

Abstract:

In this paper, we compute the scattering matrix for the Hilbert modular group over any number field $K$. We then compute the determinant of the scattering matrix and show it is a ratio of completed Dedekind zeta functions associated to the Hilbert class field of $K$. This generalizes work of Efrat and Sarnak in the imaginary quadratic case.

References

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, DOI 10.1090/S0002-9947-1985-0792829-1
Erich Hecke, Lectures on the theory of algebraic numbers, Graduate Texts in Mathematics, vol. 77, Springer-Verlag, New York-Berlin, 1981. Translated from the German by George U. Brauer, Jay R. Goldman and R. Kotzen. MR 638719
Robert P. Langlands, On the functional equations satisfied by Eisenstein series, Lecture Notes in Mathematics, Vol. 544, Springer-Verlag, Berlin-New York, 1976. MR 0579181
Peter D. Lax and Ralph S. Phillips, Scattering theory for automorphic functions, Annals of Mathematics Studies, No. 87, Princeton University Press, Princeton, N.J., 1976. MR 0562288
Jürgen Neukirch, Algebraic number theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 322, Springer-Verlag, Berlin, 1999. Translated from the 1992 German original and with a note by Norbert Schappacher; With a foreword by G. Harder. MR 1697859, DOI 10.1007/978-3-662-03983-0
R. Phillips and P. Sarnak, Perturbation theory for the Laplacian on automorphic functions, J. Amer. Math. Soc. 5 (1992), no. 1, 1–32. MR 1127079, DOI 10.1090/S0894-0347-1992-1127079-X
Andrei Reznikov, Eisenstein matrix and existence of cusp forms in rank one symmetric spaces, Geom. Funct. Anal. 3 (1993), no. 1, 79–105. MR 1204788, DOI 10.1007/BF01895514
P. Sarnak, On cusp forms. II, Festschrift in honor of I. I. Piatetski-Shapiro on the occasion of his sixtieth birthday, Part II (Ramat Aviv, 1989) Israel Math. Conf. Proc., vol. 3, Weizmann, Jerusalem, 1990, pp. 237–250. MR 1159118
Carl Ludwig Siegel, Lectures on advanced analytic number theory, Tata Institute of Fundamental Research Lectures on Mathematics, No. 23, Tata Institute of Fundamental Research, Bombay, 1965. Notes by S. Raghavan. MR 0262150
Claus Mazanti Sorensen, Fourier expansion of Eisenstein series on the Hilbert modular group and Hilbert class fields, Trans. Amer. Math. Soc. 354 (2002), no. 12, 4847–4869. MR 1926839, DOI 10.1090/S0002-9947-02-03109-4
Audrey Terras, Harmonic analysis on symmetric spaces and applications. II, Springer-Verlag, Berlin, 1988. MR 955271, DOI 10.1007/978-1-4612-3820-1
André Weil, Basic number theory, Die Grundlehren der mathematischen Wissenschaften, Band 144, Springer-Verlag New York, Inc., New York, 1967. MR 0234930