  We have searched the Kepler light curves of ~3900 M-star targets for evidence
of periodicities that indicate, by means of the effects of starspots, rapid
stellar rotation. Several analysis techniques, including Fourier transforms,
inspection of folded light curves, 'sonograms', and phase tracking of
individual modulation cycles, were applied in order to distinguish the
periodicities due to rapid rotation from those due to stellar pulsations,
eclipsing binaries, or transiting planets. We find 178 Kepler M-star targets
with rotation periods, P_rot, of < 2 days, and 110 with P_rot < 1 day. Some 30
of the 178 systems exhibit two or more independent short periods within the
same Kepler photometric aperture, while several have three or more short
periods. Adaptive optics imaging and modeling of the Kepler pixel response
function for a subset of our sample support the conclusion that the targets
with multiple periods are highly likely to be relatively young physical binary,
triple, and even quadruple M star systems. We explore in detail the one object
with four incommensurate periods all less than 1.2 days, and show that two of
the periods arise from one of a close pair of stars, while the other two arise
from the second star, which itself is probably a visual binary. If most of
these M-star systems with multiple periods turn out to be bound M stars, this
could prove a valuable way of discovering young hierarchical M-star systems;
the same approach may also be applicable to G and K stars. The ~5% occurrence
rate of rapid rotation among the ~3900 M star targets is consistent with spin
evolution models that include an initial contraction phase followed by magnetic
braking, wherein a typical M star can spend several hundred Myr before spinning
down to periods longer than 2 days.
