  We have analysed 10 months of public data from the Kepler space mission to
measure rotation periods of main-sequence stars with masses between 0.3 and
0.55 M_sun. To derive the rotational period we introduce the autocorrelation
function and show that it is robust against phase and amplitude modulation and
residual instrumental systematics. Of the 2483 stars examined, we detected
rotation periods in 1570 (63.2%), representing an increase of a factor ~ 30 in
the number of rotation period determination for field M-dwarfs. The periods
range from 0.37-69.7 days, with amplitudes ranging from 1.0-140.8 mmags. The
rotation period distribution is clearly bimodal, with peaks at ~ 19 and ~ 33
days, hinting at two distinct waves of star formation, a hypothesis that is
supported by the fact that slower rotators tend to have larger proper motions.
The two peaks of the rotation period distribution form two distinct sequences
in period-temperature space, with the period decreasing with increasing
temperature, reminiscent of the Vaughan-Preston gap. The period-mass
distribution of our sample shows no evidence of a transition at the fully
convective boundary. On the other hand, the slope of the upper envelope of the
period-mass relation changes sign around 0.55 M_sun, below which period rises
with decreasing mass.
