×

Selberg type zeta function for the Hilbert modular group of a real quadratic field. (English) Zbl 1305.11077

Summary: 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 \(\mathbb{C}\), its functional equation and some arithmetic applications.
Proofs are given in J. Number Theory 147, 396–453 (2015; Zbl 1380.11073).

MSC:

11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)
11F72 Spectral theory; trace formulas (e.g., that of Selberg)

Citations:

Zbl 1380.11073
PDFBibTeX XMLCite
Full Text: DOI Euclid

References:

[1] 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. · Zbl 1113.11055
[2] 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. · Zbl 0607.10023
[3] 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.
[4] Y. Gon, Differences of the Selberg trace formula and Selberg type zeta functions for Hilbert modular surfaces, arXiv: · Zbl 1380.11073
[5] 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. · Zbl 1236.11081
[6] D. A. Hejhal, The Selberg trace formula for \(\mathrm{PSL}(2,\mathbf{R})\). Vol. 2 , Lecture Notes in Math., 1001, Springer, Berlin, 1983. · Zbl 0543.10020
[7] P. Sarnak, Class numbers of indefinite binary quadratic forms, J. Number Theory 15 (1982), no. 2, 229-247. · Zbl 0499.10021
[8] 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. · Zbl 0072.08201
[9] A. Selberg, Collected papers. Vol. I , Springer, Berlin, 1989, pp. 626-674. · Zbl 0675.10001
[10] 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. · Zbl 0488.10027
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.