  The abundance of the faintest galaxies provides insight into the nature of
dark matter and the process of dwarf galaxy formation. In the LCDM scenario,
low mass halos are so numerous that the efficiency of dwarf formation must
decline sharply with decreasing halo mass in order to accommodate the relative
scarcity of observed dwarfs and satellites in the Local Group. The nature of
this decline contains important clues to the mechanisms regulating the onset of
galaxy formation in the faintest systems. We explore here two possible models
for the stellar mass ($M_*$)-halo mass ($M_{200}$) relation at the faint end,
motivated by some of the latest LCDM cosmological hydrodynamical simulations.
One model includes a sharp mass threshold below which no luminous galaxies
form, as expected if galaxy formation proceeds only in systems above the
Hydrogen-cooling limit. In the second model, $M_*$ scales as a steep power-law
of $M_{200}$ with no explicit cutoff, as suggested by recent semianalytic work.
Although both models predict satellite numbers around Milky Way-like galaxies
consistent with current observations, they predict vastly different numbers of
ultra-faint dwarfs and of satellites around isolated dwarf galaxies. Our
results illustrate how the satellite mass function around dwarfs may be used to
probe the $M_*$-$M_{200}$ relation at the faint end and to elucidate the
mechanisms that determine which low-mass halos "light up" or remain dark in the
LCDM scenario.
