  (Abbreviated) Particle-in-cell simulations of relativistic, weakly magnetized
collisionless shocks show that particles can gain energy by repeatedly crossing
the shock front. This requires scattering off self-generated small length-scale
magnetic fluctuations. The radiative signature of this first-order Fermi
acceleration mechanism is important for models of both the prompt and afterglow
emission in gamma-ray bursts and depends on the strength parameter "a" of the
fluctuations. For electrons (and positrons), acceleration saturates when the
radiative losses produced by the scattering cannot be compensated by the energy
gained on crossing the shock. We show that this sets an upper limit on both the
electron Lorentz factor and on the energy of the photons radiated during the
scattering process. This rules out "jitter" radiation on self-excited
fluctuations with a < 1 as a source of gamma-rays, although high-energy photons
might still be produced when the jitter photons are upscattered in an analog of
the synchrotron self-Compton process. In fluctuations with a > 1, radiation is
generated by the standard synchrotron mechanism, and the maximum photon energy
rises linearly with a, until saturating at approximately 70 MeV.
