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Transactions of the American Mathematical Society

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The determinant of the Eisenstein matrix and Hilbert class fields


Authors: I. Efrat and P. Sarnak
Journal: Trans. Amer. Math. Soc. 290 (1985), 815-824
MSC: Primary 11F41
DOI: https://doi.org/10.1090/S0002-9947-1985-0792829-1
MathSciNet review: 792829
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Abstract | References | Similar Articles | Additional Information

Abstract: We compute the determinant of the Eisenstein matrix associated to the Hilbert-Blumenthal modular group $ {\text{PSL}_2}({\mathcal{O}_k})$, and express it in terms of the zeta function of the Hilbert class field of $ K$.


References [Enhancements On Off] (What's this?)

  • [1] P. Cohen and P. Sarnak, Discrete groups and geometry (to appear).
  • [2] Isaac Y. Efrat, The Selberg trace formula for 𝑃𝑆𝐿₂(𝑅)ⁿ, Mem. Amer. Math. Soc. 65 (1987), no. 359, iv+111. MR 874084, https://doi.org/10.1090/memo/0359
  • [3] 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
  • [4] Dennis A. Hejhal, The Selberg trace formula for 𝑃𝑆𝐿(2,𝑅). Vol. 2, Lecture Notes in Mathematics, vol. 1001, Springer-Verlag, Berlin, 1983. MR 711197
  • [5] M. N. Huxley, Scattering matrices for congruence subgroups, Modular forms (Durham, 1983) Ellis Horwood Ser. Math. Appl.: Statist. Oper. Res., Horwood, Chichester, 1984, pp. 141–156. MR 803366
  • [6] Serge Lang, Algebraic number theory, Addison-Wesley Publishing Co., Inc., Reading, Mass.-London-Don Mills, Ont., 1970. MR 0282947
  • [7] Peter D. Lax and Ralph S. Phillips, Scattering theory, Pure and Applied Mathematics, Vol. 26, Academic Press, New York-London, 1967. MR 0217440
  • [8] P. Sarnak, The arithmetic and geometry of some hyperbolic three-manifolds, Acta Math. 151 (1983), no. 3-4, 253–295. MR 723012, https://doi.org/10.1007/BF02393209
  • [9] A. Terras, Harmonic analysis on symmetric spaces with applications to number theory (to appear).

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Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1985-0792829-1
Article copyright: © Copyright 1985 American Mathematical Society