Proceedings of the Japan Academy, Series A, Mathematical Sciences

Selberg type zeta function for the Hilbert modular group of a real quadratic field

Yasuro Gon

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Abstract

In this article we announce fundamental results of Selberg type zeta functions for the Hilbert modular group of a real quadratic field; the meromorphic extension over $\mathbf{C}$, its functional equation and some arithmetic applications.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 88, Number 9 (2012), 145-148.

Dates
First available in Project Euclid: 6 November 2012

Permanent link to this document
https://projecteuclid.org/euclid.pja/1352210375

Digital Object Identifier
doi:10.3792/pjaa.88.145

Mathematical Reviews number (MathSciNet)
MR3000892

Zentralblatt MATH identifier
1305.11077

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

Keywords
Hilbert modular group Selberg zeta function

Citation

Gon, Yasuro. Selberg type zeta function for the Hilbert modular group of a real quadratic field. Proc. Japan Acad. Ser. A Math. Sci. 88 (2012), no. 9, 145--148. doi:10.3792/pjaa.88.145. https://projecteuclid.org/euclid.pja/1352210375


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References

  • A. Deitmar, Generalised Selberg zeta functions and a conjectural Lefschetz formula, in Multiple Dirichlet series, automorphic forms, and analytic number theory, 177–190, Proc. Sympos. Pure Math., 75 Amer. Math. Soc., Providence, RI, 2006.
  • I. Y. Efrat, The Selberg trace formula for $\mathrm{PSL}_{2}(\mathbf{R})^{n}$, Mem. Amer. Math. Soc. 65 (1987), no. 359, iv+1–111.
  • 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.
  • Y. Gon, Differences of the Selberg trace formula and Selberg type zeta functions for Hilbert modular surfaces, arXiv:1208.6086.
  • Y. Gon and J. Park, The zeta functions of Ruelle and Selberg for hyperbolic manifolds with cusps, Math. Ann. 346 (2010), no. 3, 719–767.
  • D. A. Hejhal, The Selberg trace formula for $\mathrm{PSL}(2,\mathbf{R})$. Vol. 2, Lecture Notes in Math., 1001, Springer, Berlin, 1983.
  • P. Sarnak, Class numbers of indefinite binary quadratic forms, J. Number Theory 15 (1982), no. 2, 229–247.
  • 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.
  • A. Selberg, Collected papers. Vol. I, Springer, Berlin, 1989, pp. 626–674.
  • P. G. Zograf, Selberg trace formula for the Hilbert modular group of a real quadratic algebraic number field, J. Math. Sci. 19 (1982), no. 6, 1637–1652.