Let M be a compact Riemannian manifold. It has been known for a long time that the singularities of the wave trace, trace(cos √Δt), are located at the periods of the closed geodesics. Do these singularities also contain information about the geometry of M in the neighborhood of a closed geodesic? We prove that the Birkhoff canonical form of the Poincaré map can be determined from the singularities of the wave trace.