Determinant expression of Selberg zeta functions. I

Abstract

We will prove that for PSL(2, R) and its cofinite subgroup, the Selberg zeta function is expressed by the determinant of the Laplacian. We will also give an explicit calculation in case of congruence subgroups, and deduce that the part of the determinant of the Laplacian composed of the continuous spectrum is expressed by Dirichlet ?-functions. The first discovery of the relation between the Selberg zeta function and the determinant of the Laplacian was by physicists (3, 4, 7). Sarnak (14) and Voros ( 15) obtained the determinant expression of Selberg zeta functions for compact Riemann surfaces with torsionfree fundamental groups. In those cases, all the spectrum of the Laplacians are discrete. The determinant was defined via the holomorphy at the origin of the spectral zeta function of Minakshisundaram and Pleijel (13). For noncompact but finite Riemann surfaces with torsion- free fundamental groups, these results are generalized by Efrat (5). In this case there exist both discrete and continuous spectrum. He constructs the spectral zeta function composed not only of eigenvalues but some values concerning continuous spectrum, which are decided by all the poles of the scattering de- terminant in the Selberg trace formula. The determinant of the Laplacian is defined by the standard method with the holomorphy of the spectral zeta func- tion at the origin. The aim of the present paper is to generalize his results to the case with any fundamental group Y (§3) and to give some arithmetic examples of the determinant of the Laplacians (§4). In §4, we restrict ourselves to the case when Y is a congruence subgroup of PSL(2, Z). In this case the partial spectral zeta function composed of only eigenvalues is also holomorphic at the origin (11, Theorem 3.3). Hence we have a decomposition of the determinant into parts corresponding to the discrete and continuous spectrum. The scatter- ing determinant is expressed very explicitly by Huxley (8) in terms of Dirichlet 7-functions £(i, x) ■ Almost all the poles of the scattering determinant are de- scribed by nontrivial zeros of L(s, x) ■ The continuous part of the determinant