  We describe cosmological galaxy formation simulations with the adaptive mesh
refinement code Enzo that incorporate a star formation prescription regulated
by the local abundance of molecular hydrogen. We show that this H2-regulated
prescription leads to a suppression of star formation in low mass halos (M_h <
~10^10 M_sun) at z>4, alleviating some of the dwarf galaxy problems faced by
theoretical galaxy formation models. H2 regulation modifies the efficiency of
star formation of cold gas directly, rather than indirectly reducing the cold
gas content with "supernova feedback". We determine the local H2 abundance in
our most refined grid cells (76 proper parsec in size at z=4) by applying the
model of Krumholz, McKee, & Tumlinson, which is based on idealized 1D radiative
transfer calculations of H2 formation-dissociation balance in ~100 pc
atomic--molecular complexes. Our H2-regulated simulations are able to reproduce
the empirical (albeit lower z) Kennicutt-Schmidt relation, including the low
Sigma_gas cutoff due to the transition from atomic to molecular phase and the
metallicity dependence thereof, without the use of an explicit density
threshold in our star formation prescription. We compare the evolution of the
luminosity function, stellar mass density, and star formation rate density from
our simulations to recent observational determinations of the same at z=4-8 and
find reasonable agreement between the two.
