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Determinant expression of Selberg zeta functions. II


Author: Shin-ya Koyama
Journal: Trans. Amer. Math. Soc. 329 (1992), 755-772
MSC: Primary 11F72
DOI: https://doi.org/10.1090/S0002-9947-1992-1141858-0
MathSciNet review: 1141858
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Abstract: This paper is the $ \operatorname{PSL} (2,{\mathbf{C}})$-version of Part I. We show that for $ \operatorname{PSL} (2,{\mathbf{C}})$ and its subgroup $ \operatorname{PSL} (2,O)$, the Selberg zeta function with its gamma factors is expressed as the determinant of the Laplacians, where $ O$ is the integer ring of an imaginary quadratic field. All the gamma factors are calculated explicitly. We also give an explicit computation to the contribution of the continuous spectrum to the determinant of the Laplacian.


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  • [1] I. Efrat, Determinant of Laplacians on surfaces of finite volume, Comm. Math. Phys. 119 (1988), 443-451. MR 969211 (90c:58184)
  • [2] I. Efrat and P. Sarnak, The determinant of the Eisenstein matrix and Hilbert class fields, Trans. Amer. Math. Soc. 290 (1985), 815-824. MR 792829 (87b:11039)
  • [3] J. Elstrodt, F. Grunewald, and J. Mennicke, The Selberg zeta function for cocompact discrete subgroups of $ \operatorname{PSL} (2,{\mathbf{C}})$, Banach Center Publ., no. 17, PWN, 1985, pp. 83-120. MR 840474 (87h:11044)
  • [4] R. Gangolli and G. Warner, Zeta functions of Selberg's type for some non-compact quotients of symmetric spaces of rank one, Nagoya Math. J. 78 (1980), 1-44. MR 571435 (82m:58049)
  • [5] D. A. Hejhal, The Selberg trace formula for $ \operatorname{PSL} (2,{\mathbf{R}})$, Vol. 2, Lecture Notes in Math., vol. 1001, Springer, 1983. MR 711197 (86e:11040)
  • [6] S. Koyama, Determinant expression of Selberg zeta functions. I, Trans. Amer. Math. Soc. 324 (1991), 149-168. MR 1041049 (91h:11043)
  • [7] -, Determinant expression of Selberg zeta functions. III, Proc. Amer. Math. Soc. 113 (1991), 303-311. MR 1062391 (93h:11052)
  • [8] N. Kurokawa, Parabolic components of zeta functions, Proc. Japan Acad. Ser. A Math. Sci. 64 (1988), 21-24. MR 953756 (89m:11052)
  • [9] P. Sarnak, Determinants of Laplacians, Comm. Math. Phys. 110 (1987), 113-120. MR 885573 (89e:58116)
  • [10] G. Warner, Selberg's trace formula for nonuniform lattices: The $ {\mathbf{R}}$-rank one case, Adv. Math. Suppl. Stud. 6 (1979), 1-142. MR 535763 (81f:10044)
  • [11] E. T. Whittaker and G. N. Watson, A course of modern analysis, Cambridge Univ. Press, Cambridge, 1927. MR 1424469 (97k:01072)

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DOI: https://doi.org/10.1090/S0002-9947-1992-1141858-0
Article copyright: © Copyright 1992 American Mathematical Society

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