From July 2008 to October 2009, the Gamma-ray Burst Monitor (GBM) on board the Fermi Gamma-ray Space Telescope (FGST) has detected 320 Gamma-Ray Bursts (GRBs). About 20% of these events are classified as short based on their T90 duration below 2 s. We present here for the first time time-resolved spectroscopy at timescales as short as 2 ms for the three brightest short GRBs observed with GBM. The time-integrated spectra of the events deviate from the Band function, indicating the existence of an additional spectral component, which can be fit by a power-law with index ~-1.5. The time-integrated Epeak values exceed 2 MeV for two of the bursts, and are well above the values observed in the brightest long GRBs. Their Epeak values and their low-energy power-law indices ({alpha}) confirm that short GRBs are harder than long ones. We find that short GRBs are very similar to long ones, but with light curves contracted in time and with harder spectra stretched towards higher energies. In our time-resolved spectroscopy analysis, we find that the Epeak values range from a few tens of keV up to more than 6 MeV. In general, the hardness evolutions during the bursts follows their flux/intensity variations, similar to long bursts. However, we do not always see the Epeak leading the light-curve rises, and we confirm the zero/short average light-curve spectral lag below 1 MeV, already established for short GRBs. We also find that the time-resolved low-energy power-law indices of the Band function mostly violate the limits imposed by the synchrotron models for both slow and fast electron cooling and may require additional emission processes to explain the data. Finally, we interpreted these observations in the context of the current existing models and emission mechanisms for the prompt emission of GRBs.
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