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 _ { \odot } .
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 \sim 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 \sim 19 and \sim 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 _ { \odot } , below which period rises with decreasing mass .