Lyman Alpha line properties at $z \simeq 3.78$ and their environmental
  dependence: a case study around a massive proto-cluster
  Ly$\alpha$-emitting galaxies (LAEs) are easily detectable in the
high-redshift Universe and are potentially efficient tracers of large scale
structure at early epochs, as long as their observed properties do not strongly
depend on environment. We investigate the luminosity and equivalent width
functions of LAEs in the overdense field of a protocluster at redshift $z
\simeq 3.78$. Using a large sample of LAEs (many spectroscopically confirmed),
we find that the Ly$\alpha$ luminosity distribution is well-represented by a
Schechter (1976) function with $\log(L^{\ast}/{\rm erg s^{-1}}) =
43.26^{+0.20}_{-0.22}$ and $\log(\phi^{\ast}/{\rm
Mpc^{-3}})=-3.40^{+0.03}_{-0.04}$ with $\alpha=-1.5$. Fitting the equivalent
width distribution as an exponential, we find a scale factor of
$\omega=79^{+15}_{-15}$ Angstroms. We also measured the Ly$\alpha$ luminosity
and equivalent width functions using the subset of LAEs lying within the
densest cores of the protocluster, finding similar values for $L^*$ and
$\omega$. Hence, despite having a mean overdensity more than 2$\times$ that of
the general field, the shape of the Ly$\alpha$ luminosity function and
equivalent width distributions in the protocluster region are comparable to
those measured in the field LAE population by other studies at similar
redshift. While the observed Ly$\alpha$ luminosities and equivalent widths show
correlations with the UV continuum luminosity in this LAE sample, we find that
these are likely due to selection biases and are consistent with no intrinsic
correlations within the sample. This protocluster sample supports the strong
evolutionary trend observed in the Ly$\alpha$ escape fraction and suggest that
lower redshift LAEs can be on average significantly more dusty that their
counterparts at higher redshift.
