  The massive scalar field theory and the chiral Schwinger model are quantized
on a Poincar\'e disk of radius $\rho$. The amplitudes are derived in terms of
hypergeometric functions. The behavior at long distances and near the boundary
of some of the relevant correlation functions is studied. The exact computation
of the chiral determinant appearing in the Schwinger model is obtained
exploiting perturbation theory. This calculation poses interesting mathematical
problems, as the Poincar\'e disk is a noncompact manifold with a metric tensor
which diverges approaching the boundary. The results presented in this paper
are very useful in view of possible extensions to general Riemann surfaces.
Moreover, they could also shed some light in the quantization of field theories
on manifolds with constant curvature scalars in higher dimensions.
