  We use $N$-body simulations to show that high-redshift galaxy counts provide
an interesting constraint on the nature of dark matter, specifically Warm Dark
Matter (WDM), owing to the lack of early structure formation these models. Our
simulations include three WDM models with thermal-production masses of 0.8 keV,
1.3 keV, and 2.6 keV, as well as CDM. Assuming a relationship between dark halo
mass and galaxy luminosity that is set by the observed luminosity function at
bright magnitudes, we find that 0.8 keV WDM is disfavored by direct galaxy
counts in the Hubble Ultra Deep Field at $>\!\!10\sigma$. Similarly, 1.3 keV
WDM is statistically inconsistent at $2.2\sigma$. Future observations with JWST
(and possibly HST via the Frontier Fields) could rule out $1.3$ keV WDM at high
significance, and may be sensitive to WDM masses greater than 2.6 keV. We also
examine the ability of galaxies in these WDM models to reionize the universe,
and find that 0.8 keV and 1.3 keV WDM produce optical depths to the Cosmic
Microwave Background (CMB) that are inconsistent at 68% C.L. with current
Planck results, even with extremely high ionizing radiation escape fractions,
and 2.6 keV WDM requires an optimistic escape fraction to yield an optical
depth consistent with Planck data. Although CMB optical depth calculations are
model dependent, we find a strong challenge for stellar processes alone to
reionize the universe in a 0.8 keV and 1.3 keV WDM cosmology.
