The Selberg zeta function and the determinant of the Laplacians (Q913847)

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.