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

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

2004-10-04

arxiv.org/abs/math/0410067v2

Mathematics

Number Theory

25 pages, submitted to Mathematische Zeitschrift

Scientific paper

10.1007/s00209-005-0806-9

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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