  Context. More than one hundred GeV pulsars have been detected up to now by
the LAT telescope on the Fermi gamma-ray observatory, showing peak energies
around a few GeV. Current modelling proposes that the high energy emission
comes from outer magnetospheric gaps, however radiation from the equatorial
current sheet which separates the two magnetic hemispheres outside the light
cylinder has also been investigated. Aims. In this paper we discuss the region
right outside the light cylinder, or "near wind" zone. We investigate the
possibility that synchrotron radiation emitted by thermal populations in the
equatorial current sheet of the pulsar wind in this region can explain the
lightcurves and spectra observed by Fermi/LAT. Methods. We use analytical
estimates as well as detailed numerical computation to calculate the gamma-ray
luminosities, lightcurves and spectra of gamma-ray pulsars. Results. Many of
the characteristics of the gamma-ray pulsars observed by Fermi/LAT can be
reproduced by our model, most notably the position of these objects in the P -
Pdot diagram, and the range of gamma-ray luminosities. A testable result is a
sub-exponential cutoff with an index b = 0.35. We also predict the existence of
a population of pulsars with cutoff energies in the MeV range. These have
systematically lower spindown luminosities than the Fermi/LAT detected pulsars.
Conclusions. It is possible for relativistic populations of electrons and
positrons in the current sheet of a pulsar's wind right outside the light
cylinder to emit synchrotron radiation that peaks in the sub-GeV to GeV regime,
with gamma-ray efficiencies similar to those observed for the Fermi/LAT
pulsars.
