Artin formalism for Selberg zeta functions of co-finite Kleinian groups
Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 1, p. 59-75

Let Γ 3 be a finite-volume quotient of the upper-half space, where Γ SL (2,) 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,), 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.

Soit Γ un sous-groupe discret de SL (2,) tel que le quotient Γ 3 ait un volume fini. On associe à une représentation unitaire de dimension finie χ de Γ la fonction zêta de Selberg Z(s;Γ;χ). Dans cet article, on prouve le formalisme d’Artin pour cette fonction zêta de Selberg. Plus précisément, si Γ ˜ est une extension de Γ d’indice fini dans SL (2,), et si π= Ind Γ Γ ˜ χ est la représentation induite, alors Z(s;Γ;χ)=Z(s;Γ ˜;π). Dans la deuxième partie de l’article, on prouve par une méthode directe l’identité analogue pour la fonction de dispersion. Plus précisément, φ(s;Γ;χ)=φ(s;Γ ˜;π) pour une certaine normalisation de la série d’Eisenstein.

DOI : https://doi.org/10.5802/jtnb.657
Keywords: Artin Formalism, Selberg Zeta function, Kleinian groups, Fuchsian groups hyperbolic 3-manifolds, scattering matrix, Eisenstein series.
@article{JTNB_2009__21_1_59_0,
     author = {Brenner, Eliot and Spinu, Florin},
     title = {Artin formalism for Selberg zeta functions of co-finite Kleinian groups},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {21},
     number = {1},
     year = {2009},
     pages = {59-75},
     doi = {10.5802/jtnb.657},
     mrnumber = {2537703},
     zbl = {pre05620668},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2009__21_1_59_0}
}
Brenner, Eliot; Spinu, Florin. Artin formalism for Selberg zeta functions of co-finite Kleinian groups. Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 1, pp. 59-75. doi : 10.5802/jtnb.657. http://www.numdam.org/item/JTNB_2009__21_1_59_0/

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