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 0.55M_{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 of ~30 in the number of rotation period determination for field M dwarfs. The periods range from 0.37 to 69.7d, with amplitudes ranging from 1.0 to 140.8mmag. The rotation period distribution is clearly bimodal, with peaks at ~19 and ~33d, 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 0.55M{sun}_, below which period rises with decreasing mass.

Cone search capability for table J/MNRAS/432/1203/table2 (M-dwarfs with detected rotation periods)

Cone search capability for table J/MNRAS/432/1203/table3 (Objects identified as likely giants)

Cone search capability for table J/MNRAS/432/1203/table4 (Objects with 2 distinct periods)

Cone search capability for table J/MNRAS/432/1203/table5 (Objects with no rotation period detection)