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

Friedman, Joshua S.

doi:10.1090/s0002-9939-08-09516-6

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2009

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“…The proof of the identity of Selberg zeta functions presented in this paper is very similar to the one in [5], which studied independently the special case when Γ is a normal subgroup ofΓ. The proof differs from previous approaches in that it avoids entirely the trace formula.…”

Section: Introductionmentioning

confidence: 70%

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

Brenner¹,

Spinu²

2009

J. Théor. Nombres Bordeaux

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Let Γ\H 3 be a finite-volume quotient of the upper-half space, where Γ ⊂ SL(2, C) is a discrete subgroup. To a finite dimensional unitary representation χ of Γ one associates the Selberg zeta function Z(s; Γ; χ). In this paper we prove the Artin formalism for the Selberg zeta function. Namely, if Γ is a finite index group extension of Γ in SL(2, C), and π = Ind Γ Γ χ is the induced representation, then Z(s; Γ; χ) = Z(s; Γ; π). In the second part of the paper we prove by a direct method the analogous identity for the scattering function, namely φ(s; Γ; χ) = φ(s; Γ; π), for an appropriate normalization of the Eisenstein series.

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Section: Introductionmentioning

confidence: 70%