We characterize the subscheme of the moduli space of torsion-free sheaves on
an elliptic surface which are stable of relative degree zeero (with respect to
a polarization of type aH+bf, H being the section and f the elliptic fibre)
which is isomorphic, via the relative Fourier-Mukai transform, with the
relative compactified Simpson Jacobian of the family of those curves D in the
surface which are flat over the base of the elliptic fibration. This
generalizes and completes earlier constructions due to Friedman, Morgan and
Witten. We also study the relative moduli scheme of sheaves whose restriction
to each fibre is torsion-free and semistable of rank n and degree zero for
higher dimensional elliptic fibrations. The relative Fourier-Mukai transform
induces an isomorphic between this relative moduli space and the relative n-th
symmetric product of the fibration.Comment: AMS-LaTeX, 18 pages, XY-pic; new title, some modifications; final
version as accepted in J. Geom. Phy