  We investigate the evolution of the faint-end slope of the luminosity
function, $\alpha$, using semi-analytical modeling of galaxy formation. In
agreement with observations, we find that the slope can be fitted well by
$\alpha (z) =a+b z$, with a=-1.13 and b=-0.1. The main driver for the evolution
in $\alpha$ is the evolution in the underlying dark matter mass function.
Sub-L_* galaxies reside in dark matter halos that occupy a different part of
the mass function. At high redshifts, this part of the mass function is steeper
than at low redshifts and hence $\alpha$ is steeper. Supernova feedback in
general causes the same relative flattening with respect to the dark matter
mass function. The faint-end slope at low redshifts is dominated by field
galaxies and at high redshifts by cluster galaxies. The evolution of
$\alpha(z)$ in each of these environments is different, with field galaxies
having a slope b=-0.14 and cluster galaxies b=-0.05. The transition from
cluster-dominated to field-dominated faint-end slope occurs roughly at a
redshift $z_* \sim 2$, and suggests that a single linear fit to the overall
evolution of $\alpha(z)$ might not be appropriate. Furthermore, this result
indicates that tidal disruption of dwarf galaxies in clusters cannot play a
significant role in explaining the evolution of $\alpha(z)$ at z< z_*. In
addition we find that different star formation efficiencies a_* in the
Schmidt-Kennicutt-law and supernovae-feedback efficiencies $\epsilon$ generally
do not strongly influence the evolution of $\alpha(z)$.
