Tohoku Mathematical Journal

Splitting density for lifting about discrete groups

Yasufumi Hashimoto and Masato Wakayama

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Abstract

We study splitting densities of primitive elements of a discrete subgroup of a connected non-compact semisimple Lie group of real rank one with finite center in another larger such discrete subgroup. When the corresponding cover of such a locally symmetric negatively curved Riemannian manifold is regular, the densities can be easily obtained from the results due to Sarnak or Sunada. Our main interest is a case where the covering is not necessarily regular. Specifically, for the case of the modular group and its congruence subgroups, we determine the splitting densities explicitly. As an application, we study analytic properties of the zeta function defined by the Euler product over elements consisting of all primitive elements which satisfy a certain splitting law for a given lifting.

Article information

Source
Tohoku Math. J. (2) Volume 59, Number 4 (2007), 527-545.

Dates
First available in Project Euclid: 6 January 2008

Permanent link to this document
https://projecteuclid.org/euclid.tmj/1199649873

Digital Object Identifier
doi:10.2748/tmj/1199649873

Mathematical Reviews number (MathSciNet)
MR2404204

Zentralblatt MATH identifier
1148.11026

Subjects
Primary: 11M36: Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. Explicit formulas
Secondary: 11F72: Spectral theory; Selberg trace formula

Keywords
Prime geodesic theorem splitting density Selberg's zeta function regular cover congruence subgroup semisimple Lie groups

Citation

Hashimoto, Yasufumi; Wakayama, Masato. Splitting density for lifting about discrete groups. Tohoku Math. J. (2) 59 (2007), no. 4, 527--545. doi:10.2748/tmj/1199649873. https://projecteuclid.org/euclid.tmj/1199649873.


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References

  • E. Artin, Über die Zetafunktionen gewisser algebraischer Zahlkörper, Math. Ann. 89 (1923), 147--156.
  • L. E. Dickson, Linear groups: With an exposition of the Galois field theory, Dover Phoenix Editions, Dover Publications, Inc., New York, 1958.
  • R. Gangolli and G. Warner, Zeta functions of Selberg's type for some noncompact quotients of symmetric spaces of rank one, Nagoya Math. J. 78 (1980), 1--44.
  • C. F. Gauss, Disquisitiones arithmeticae, Fleischer, Leipzig, 1801.
  • Y. Hashimoto, Arithmetic expressions of Selberg's zeta functions for congruence subgroups, J. Number Theory 122 (2007), 324--335.
  • D. Hejhal, The Selberg trace formula of $\mathrmPSL(2,\boldsymbol R)$ I, Lecture Notes in Math. 548, Springer-Verlag, Berlin, 1976/ II, Lecture Notes in Math. 1001, Springer-Verlag, Berlin, 1983.
  • H. D. Kloosterman, The behavior of general theta functions under the modular group and the characters of binary modular congruence group, I, Ann. of Math. (2) 47 (1946), 317--375.
  • W. Narkiewicz, Elementary and analytic theory of algebraic numbers, second edition, Springer-Verlag, Berlin; PWN, Warsaw, 1990.
  • P. Sarnak, Class numbers of indefinite binary quadratic forms, J. Number Theory 15 (1982), 229--247.
  • A. Selberg, Harmonic analysis, Göttingen Lecture Notes (1954), Collected papers of A. Selberg vol.1, 626--674, Springer-Verlag, Berlin, 1989.
  • T. Sunada, Fundamental groups and Laplacians, Kinokuniya Shoten, Tokyo, 1988 (in Japanese).
  • T. Sunada, $L$-functions in geometry and some applications, Curvature and topology of Riemannian manifolds (Katata, 1985), 266--284, Lecture Notes in Math. 1201, Springer, Berlin, 1986.
  • T. Takagi, Über eine theorie des relativ abelschen Zahlkörpers, Journal of the College of Science, Imperial University of Tokyo 41 (1920), 1--133.
  • T. Takagi, Algebraic number theory, Second edition, Iwanami Shoten, Tokyo, 1971 (in Japanese).
  • N. Tchebotarev, Die Bestimmung der Dichtigkeit einer Menge von Primzahlen, welch zu einer gegebenen Substitutionsklasse gehoren, Math. Ann. 95 (1926), 191--228.
  • A. B. Venkov and P. G. Zograf, Analogues of Artin's factorization formulas in the spectral theory of automorphic functions associated with induced representations of Fuchsian groups, Math. USSR Izv. 21 (1983), 435--443.