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

2004/10/19 by Joshua S. Friedman
Mathematics · #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Analytic Number Theory Research #math.NT #math.SP #msc:11F72 #msc:11M36

paper · pdf · doi:10.1007/s00209-005-0806-9

25 pages, submitted to Mathematische Zeitschrift

arxiv created 2004/10/19 · openalex publication_date 2005/05/30 · arxiv updated 2009/12/01 · openalex created_date 2016/06/24 · openalex updated_date 2026/07/29

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, if D is the ring of Eisenstein integers, then the Selberg zeta-function of PSL(2,D) contains ramification points and is the sixth-root of a meromorphic function.

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