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

Eliot Brenner; Florin Spinu

doi:10.5802/jtnb.657

Eliot Brenner ¹; Florin Spinu ²

¹ University of Minnesota 206 Church Street SE Minneapolis USA, 55455
² Johns Hopkins University Baltimore USA, 21218

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

Abstract
Résumé

Let $Γ ∖ ℍ^{3}$ be a finite-volume quotient of the upper-half space, where $Γ \subset SL (2, ℂ)$ 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 $\tilde{Γ}$ is a finite index group extension of $Γ$ in $SL (2, ℂ)$ , and $π = {Ind}_{Γ}^{\tilde{Γ}} χ$ is the induced representation, then $Z (s; Γ; χ) = Z (s; \tilde{Γ}; π)$ . In the second part of the paper we prove by a direct method the analogous identity for the scattering function, namely $φ (s; Γ; χ) = φ (s; \tilde{Γ}; π)$ , for an appropriate normalization of the Eisenstein series.

Soit $Γ$ un sous-groupe discret de $SL (2, ℂ)$ tel que le quotient $Γ ∖ ℍ^{3}$ 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 $\tilde{Γ}$ est une extension de $Γ$ d’indice fini dans $SL (2, ℂ)$ , et si $π = {Ind}_{Γ}^{\tilde{Γ}} χ$ est la représentation induite, alors $Z (s; Γ; χ) = Z (s; \tilde{Γ}; π)$ . Dans la deuxième partie de l’article, on prouve par une méthode directe l’identité analogue pour la fonction de dispersion. Plus précisément, $φ (s; Γ; χ) = φ (s; \tilde{Γ}; π)$ pour une certaine normalisation de la série d’Eisenstein.

MR

DOI: 10.5802/jtnb.657

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

Author's affiliations:

Eliot Brenner ¹; Florin Spinu ²

¹ University of Minnesota 206 Church Street SE Minneapolis USA, 55455
² Johns Hopkins University Baltimore USA, 21218

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     title = {Artin formalism for {Selberg} zeta functions of co-finite {Kleinian} groups},
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Eliot Brenner; Florin Spinu. 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. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.657/

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