The Selberg trace formula and Selberg zeta-function for cofinite Kleinian groups with finite-dimensional unitary representations

Abstract

ABSTRACT. For cofinite Kleinian groups, with finite-dimensional unitary representations, we derive the Selberg trace formula. As an application we define the corresponding Selberg zeta-function and compute its divisor, thus generalizing results of Elstrodt, Grunewald and Mennicke to non-trivial unitary representations. We show that the presence of cuspidal elliptic elements sometimes adds ramification point to the zeta function. In fact, if2 ] is the ring of Eisenstein integers, then the Selberg zeta-function of PSL(2, O) contains ramification points and is the sixth-root of a meromorphic function.

Artin formalism for Selberg zeta functions of co-finite Kleinian groups

“…Associated to χ, the Selberg zeta function Z(s; Γ; χ) is a function of the complex parameter s usually given by an infinite product analogous to the Euler factorization of the Artin L-functions ( [8,3,4,10]). Its logarithmic derivative has a simpler expression and we will work with that instead.…”

Analogues of the Artin factorization formula for the automorphic scattering matrix and Selberg zeta-function associated to a Kleinian group

“…We allow Γ to be an arbitrary Kleinian group. See [Fri05a] and [EGM98, Sections 5.2,5.4] for the details on its construction.…”

“…We will not need the full trace formula found in [Fri05a,Fri05b]. Rather, only parts of its proof will be needed.…”

Regularized Determinants of the Laplacian for cofinite kleinian groups with Finite-Dimensional Unitary Representations

“…The main tool that allows us to study the asymptotic behavior (near t = 0 and t = ∞) of the regularized heat kernel is the Selberg trace formula for the case of a cofinite Kleinian group with finite-dimensional unitary representations [Fri05a,Fri05b].…”

“…Or in other words the Laplacian acting on the Hilbert space of χ−automorphic (χ is a finite-dimensional unitary representation) functions on hyperbolic three-space. We relate the determinant to the Selberg zeta-function using the appropriate version of the Selberg trace formula (proved previously in [Fri05a,Fri05b]). …”