We have analyzed Kepler light curves for 849 stars with T_eff < 5200 K from
our Cycle 1 Guest Observer program. We identify six new eclipsing binaries, one
of which has an orbital period of 29.91 d, and two of which are probably W UMa
variables. In addition, we identify a candidate "warm Jupiter" exoplanet. We
further examine a subset of 670 sources for variability. Of these objects, 265
stars clearly show periodic variability that we assign to rotation of the
low-mass star. At the photometric precision level provided by Kepler, 251 of
our objects showed no evidence for variability. We were unable to determine
periods for 154 variable objects. We find that 79% of stars with T_eff < 5200 K
are variable. The rotation periods we derive for the periodic variables span
the range 0.31 < P_rot < 126.5 d. A considerable number of stars with rotation
periods similar to the solar value show activity levels that are 100 times
higher than the Sun. This is consistent with results for solar-like field
stars. As has been found in previous studies, stars with shorter rotation
periods generally exhibit larger modulations. This trend flattens beyond P_rot
= 25 d, demonstrating that even long period binaries may still have components
with high levels of activity and investigating whether the masses and radii of
the stellar components in these systems are consistent with stellar models
could remain problematic. Surprisingly, our modeling of the light curves
suggests that the active regions on these cool stars are either preferentially
located near the rotational poles, or that there are two spot groups located at
lower latitudes, but in opposing hemispheres.Comment: 48 pages, 11 figure