  The location of M dwarfs in the V-K_s--M_Ks color-magnitude diagram (CMD) has
been shown to correlate with metallicity. We demonstrate that previous
empirical photometric calibrations of M dwarf metallicity exploiting this
correlation systematically underestimate or overestimate metallicity at the
extremes of their range. We improve upon previous calibrations in three ways.
We use both a volume-limited and kinematically-matched sample of F and G dwarfs
from the Geneva-Copehnagen Survey (GCS) to infer the mean metallicity of M
dwarfs in the Solar Neighborhood, we use theoretical models of M dwarf
interiors and atmospheres to determine the effect of metallicity on M dwarfs in
the V-K_s--M_Ks CMD, and we base our final calibration purely on
high-resolution spectroscopy of FGK primaries with M dwarf companions. As a
result, we explain an order of magnitude more of the variance in the
calibration sample than previous photometric calibrations. We
non-parametrically quantify the significance of the observation that M dwarfs
that host exoplanets are preferentially in a region of the V-K_s--M_Ks plane
populated by metal-rich M dwarfs. We find that the probability p that
planet-hosting M dwarfs are distributed across the V-K_s--M_Ks CMD in the same
way as field M dwarfs is p = 0.06 +/- 0.008. Interestingly, the subsample of M
dwarfs that host Neptune and sub-Neptune mass planets may also be
preferentially located in the region of the V-K_s--M_Ks plane populated by
high-metallicity M dwarfs. The probability of this occurrence by chance is p =
0.40 +/- 0.02, and this observation hints that low-mass planets may be more
likely to be found around metal-rich M dwarfs. An increased rate of low-mass
planet occurrence around metal-rich M dwarfs would be a natural consequence of
the core-accretion model of planet formation. (abridged)
