Selberg’s zeta functions and surfaces of finite geometry. (Fonctions zêta de Selberg et surfaces de géométrie finie.) (French) Zbl 0794.58044

Guillopé, Laurent

Kurokawa, N. (ed.) et al., Zeta functions in geometry. Tokyo: Kinokuniya Company Ltd.. Adv. Stud. Pure Math. 21, 33-70 (1992).

Summary: Let \(M\) be a Riemann surface of constant curvature \(-1\), finite geometry and totally geodesic compact boundary. With a similar definition to the Selberg’s zeta function associated to a Riemann surface of finite area, the zeta function \(Z_ M\) is expressed through spectral invariants (eigenvalues and resonances) and extends so to a meromorphic function on the entire complex plane. Linked to trace formulas of Selberg’s and Birman-Krejn’s type, the proof is based on the meromorphic extensions of the resolvent of various Laplacians and the following (stationary and non-stationary) scattering theory.
For the entire collection see [Zbl 0771.00036].

MSC:

58J50	Spectral problems; spectral geometry; scattering theory on manifolds
11F72	Spectral theory; trace formulas (e.g., that of Selberg)

Keywords:

hyperbolic Riemann surface; Selberg’s zeta function; spectral invariants; meromorphic extensions