Selberg zeta-functions for cofinite lattices acting on line bundles over complex hyperbolic spaces
For a homogeneous bundle \(E\) on a locally symmetric space of rank one \(Y=\Gamma\backslash G/K\) (\(G\) semi-simple Lie group of \(\mathbb R\)-rank one, \(K\) compact maximal subgroup of \(G\) and \(\Gamma\) discrete subgroup of \(G\) without elliptic elements), the Selberg zeta-function \(Z_{\Gamma, E}\) encodes the holonomy spectrum of its periodic geodesics. The case with \(Y\) compact is rather well understood, either by the dynamical approach of \textit{D. Fried} [Ann. Sci. Éc. Norm. Supér. (4) 19, 491--517 (1986; Zbl 0609.58033)] or by harmonic analysis through a trace formula [e. g. \textit{U. Bunke} and \textit{M. Olbrich}, Selberg zeta and theta functions. A differential operator approach. Mathematical Research 83 (Akademie Verlag, Berlin) (1995; Zbl 0831.58001)]. For \(Y\) non-compact but of finite volume, the dynamical approach failed because of the non-uniform dynamical hyperbolicity in the cusps: nevertheless a trace formula does exist as proved by \textit{G. Warner} [Adv. Math. Suppl. Stud. 6, 1--142 (1979; Zbl 0466.10018)]. Its terms induced by the continuous spectrum require a special analysis, which has been done only in a few cases, e.g. for \(4d\)-dimensional complex hyperbolic manifolds by \textit{R. Gangolli} and \textit{G. Warner} [J. Math. Soc. Japan 27, 328--343 (1975; Zbl 0325.22014)]. The author considers here the case of some homogeneous line bundle \(E_\ell\) on any complex hyperbolic space : \(G=SU(n,1)\) and the bundle \(E_\ell\) is given by the \(1\)-dimensional \(\tau_\ell\) representation defined by \(\tau_\ell(\sigma,z)=z^\ell, (\sigma,z)\in K=S(U(n)\times U(1))\). Using the harmonic analysis tools established by \textit{P. C. Trombi} for \(\tau_\ell\)-spherical functions [Pac. J. Math. 101, 223--245 (1982; Zbl 0572.22005)], the author writes down the trace formula for an appropriate test function depending on a parameter \(s\). A Selberg zeta-function \(Z_{\gamma,\ell}\) is defined through its logarithmic derivative on the half-plane \(\Re s >2n\): it corresponds in the trace formula to terms indexed by the hyperbolic elements of \(\Gamma\). The author analyzes the meromorphic properties of all the remaining terms of the trace formula, which correspond to the continuous spectrum and the parabolic classes of \(\Gamma\). The analytic continuation of \(Z_{\gamma,\ell}\) follows, as well the description of its divisor and a functional equation. The function \(Z_{\gamma,\ell}\) has an Eulerian product formula taken on the set of closed geodesics of \(Y\), with an extra factor if \(| \ell| > n\) which is holomorphic on the slit plane \(\mathbb C\). The trivial case \(\ell=0\) gives the Gangolli-Warner's unsolved \(4d+2\)-dimensional case.
scientific article; zbMATH DE number 2165948