Trace formula for Riemann surfaces with magnetic field

The Maass operator for automorphic forms is interpreted as the Hamiltonian of a charged particle moving in a constant magnetic field on a Riemann surface of constant negative curvature. A careful analysis of the classical trajectories in the presence of the magnetic field shows that the action on a periodic orbit is related to the action on a free periodic geodesic. This identification provides a semiclassical interpretation of the Selberg-Maass trace formula for automorphic forms.