The Selberg zeta function and the determinant of the Laplacians
type: Item
instance of: scholarly article
title: The Selberg zeta function and the determinant of the Laplacians
zbMATH Open document ID: 0701.11031
author: Shin-ya Koyama
published in: Proceedings of the Japan Academy. Series A
publication date: 1989-01-01T00:00:00Z
review text: Let G be a noncompact connected semi-simple Lie group of rank 1 with finite center, K a maximal compact subgroup and \(\Gamma\) a cofinite discrete subgroup. In certain cases it is known that the Selberg zeta function of \(\Gamma\) has a simple relation with the determinant of the Laplacian on \(\Gamma\setminus G/K\), and the corresponding local factors are explicitly known in these cases. The author has generalized this type of determinant expression to more general G and \(\Gamma\) ; in particular, the case of non-cocompact quotients is considered. Special examples are \(G=SL_ 2({\mathbb{R}})\), \(\Gamma =\Gamma_ 0(N)\), \(\Gamma_ 1(N)\) or \(\Gamma\) (N) and \(G=SL_ 2({\mathbb{C}})\), \(\Gamma =SL_ 2({\mathfrak v})\), where \({\mathfrak v}\) is the ring of integers of an imaginary quadratic number field.
Mathematics Subject Classification ID: 11M41
Mathematics Subject Classification ID: 11F72
Mathematics Subject Classification ID: 43A85
zbMATH DE Number: 4148194
zbMATH Keywords: spectral zeta function
zbMATH Keywords: continuous spectrum
zbMATH Keywords: Selberg zeta function
zbMATH Keywords: determinant of the Laplacian
zbMATH Keywords: local factors
zbMATH Keywords: non-cocompact quotients
MaRDI profile type: Publication
full work available at URL: pjaa.65.280
OpenAlex ID: W2075813452
cites work: Zeta functions of Selberg's type for compact space forms of symmetric spaces of rank one
cites work: Determinant Expression of Selberg Zeta Functions. I
cites work: Determinant Expression of Selberg Zeta Functions. II
cites work: Determinant Expression of Selberg Zeta Functions (III)
cites work: Parabolic components of zeta functions
DOI: 10.3792/PJAA.65.280
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description: scientific article; zbMATH DE number 4148194