  In this paper, the massless Schwinger model or two dimensional quantum
electrodynamics is exactly solved on a Riemann surface. The partition function
and the generating functional of the correlation functions involving the
fermionic currents are explicitly derived using a method of quantization valid
for any abelian gauge field theory and explained in the recent references [F.
Ferrari, {\it Class. Quantum Grav.} {\bf 10} (1993), 1065], [F. Ferrari, hep-th
9310024]. In this sense, the Schwinger model is one of the few examples of
interacting and nontopological field theories that are possible to quantize on
a Riemann surface. It is also shown here that the Schwinger model is equivalent
to a nonlocal integrable model which represents a generalization of the
Thirring model. Apart from the possible applications in string theory and
integrable models, we hope that this result can be also useful in the study of
quantum field theories in curved space-times.
