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

Brenner, Eliot; Spinu, Florin

Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 1, pp. 59-75.

Let $\Gamma \setminus {ℍ}^3$ be a finite-volume quotient of the upper-half space, where $\Gamma \subset \mathrm{SL}(2,ℂ)$ is a discrete subgroup. To a finite dimensional unitary representation $\chi$ of $\Gamma$ one associates the Selberg zeta function $Z(s;\Gamma ;\chi )$. In this paper we prove the Artin formalism for the Selberg zeta function. Namely, if $\tilde{\Gamma }$ is a finite index group extension of $\Gamma$ in $\mathrm{SL}(2,ℂ)$, and $\pi ={\mathrm{Ind}}_{\Gamma }^{\tilde{\Gamma }}\chi$ is the induced representation, then $Z(s;\Gamma ;\chi )=Z(s;\tilde{\Gamma };\pi )$. In the second part of the paper we prove by a direct method the analogous identity for the scattering function, namely $\phi (s;\Gamma ;\chi )=\phi (s;\tilde{\Gamma };\pi )$, for an appropriate normalization of the Eisenstein series.

Soit $\Gamma$ un sous-groupe discret de $\mathrm{SL}(2,ℂ)$ tel que le quotient $\Gamma \setminus {ℍ}^3$ ait un volume fini. On associe à une représentation unitaire de dimension finie $\chi$ de $\Gamma$ la fonction zêta de Selberg $Z(s;\Gamma ;\chi )$. Dans cet article, on prouve le formalisme d’Artin pour cette fonction zêta de Selberg. Plus précisément, si $\tilde{\Gamma }$ est une extension de $\Gamma$ d’indice fini dans $\mathrm{SL}(2,ℂ)$, et si $\pi ={\mathrm{Ind}}_{\Gamma }^{\tilde{\Gamma }}\chi$ est la représentation induite, alors $Z(s;\Gamma ;\chi )=Z(s;\tilde{\Gamma };\pi )$. Dans la deuxième partie de l’article, on prouve par une méthode directe l’identité analogue pour la fonction de dispersion. Plus précisément, $\phi (s;\Gamma ;\chi )=\phi (s;\tilde{\Gamma };\pi )$ pour une certaine normalisation de la série d’Eisenstein.

Keywords: Artin Formalism, Selberg Zeta function, Kleinian groups, Fuchsian groups hyperbolic 3-manifolds, scattering matrix, Eisenstein series.

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