A class of automorphic functions associated with quadratic orbits of PSL(2,Z) in P1(R)

A quadratic orbit of PSL(2, Z) in P-1(R) is characterized by a quadratic irrational rho, the conjugate of which in the field Q(rho) is denoted as <(rho)over bar>. Let Sigma be the union of the locally finite set of PSL(2, Z)-transforms of the straight line (in the sense of hyperbolic geometry) from rho to <(rho)over bar> For every nu is an element of C such that Re nu < -1, we construct a certain continuous automorphic function f(nu), C-infinity in the complementary of Sigma, a solution of the equation (Delta - 1-nu(2)/4) f(nu) = 0 in this open set. The function S, can be continued as a meromorphic function of nu in the whole complex plane, with poles at the points i lambda(k) with (1 + lambda k(2))/4 in the discrete spectrum of the modular Laplacian, and at zeros of the zeta function: one expresses f(nu) in terms of the automorphic Green function. (C) 2000 Academie des sciences/Editions scientifiques et medicales Elsevier SAS.