  We review the definition of the Casimir energy steming naturally from the
concept of functional determinant through the zeta function prescription. This
is done by considering the theory at finite temperature and by defining then
the Casimir energy as its energy in the limit $T\to 0$. The ambiguity in the
coefficient $C_{d/2}$ is understood to be a result of the necessary
renormalization of the free energy of the system. Then, as an exact, explicit
example never calculated before, the Casimir energy for a massive scalar field
living in a general $(1+2)$-dimensional toroidal spacetime (i.e., a general
surface of genus one) with flat spatial geometry ---parametrized by the
corresponding Teichm\"uller parameters--- and its precise dependence on these
parameters and on the mass of the field is obtained under the form of an
analytic function.
