The rotation and Galactic kinematics of mid M dwarfs in the Solar
  Neighborhood
  Rotation is a directly-observable stellar property, and drives magnetic field
generation and activity through a magnetic dynamo. Main sequence stars with
masses below approximately 0.35Msun (mid-to-late M dwarfs) are
fully-convective, and are expected to have a different type of dynamo mechanism
than solar-type stars. Measurements of their rotation rates provide insights
into these mechanisms, but few rotation periods are available for these stars
at field ages. Using photometry from the MEarth transit survey, we measure
rotation periods for 387 nearby, mid-to-late M dwarfs in the Northern
hemisphere, finding periods from 0.1 to 140 days. The typical detected rotator
has stable, sinusoidal photometric modulations at a semi-amplitude of 0.5 to
1%. We find no period-amplitude relation for stars below 0.25Msun and an
anti-correlation between period and amplitude for higher-mass M dwarfs. We
highlight the existence of older, slowly-rotating stars without H{\alpha}
emission that nevertheless have strong photometric variability. The Galactic
kinematics of our sample is consistent with the local population of G and K
dwarfs, and rotators have metallicities characteristic of the Solar
Neighborhood. We use the W space velocities and established age-velocity
relations to estimate that stars with P<10 days are on average <2 Gyrs, and
that those with P>70 days are about 5 Gyrs. The period distribution is mass
dependent: as the mass decreases, the slowest rotators at a given mass have
longer periods, and the fastest rotators have shorter periods. We find a lack
of stars with intermediate rotation periods. [Abridged]
