TRACE FORMULA FOR RIEMANN SURFACES WITH MAGNETIC-FIELD

The Maass operator for automorphic forms is interpreted as the Hamilto nian of a charged particle moving in a constant magnetic field on a Ri emann surface of constant negative curvature. A careful analysis of th e classical trajectories in the presence of the magnetic field shows t hat the action on a periodic orbit is related to the action on a free periodic geodesic. This identification provides a semiclassical interp retation of the Selberg-Maass trace formula for automorphic forms.