  We search for evidence of diffuse Ly-alpha emission from extended neutral
hydrogen surrounding Ly-alpha emitting galaxies (LAEs) using deep narrow-band
images of the Extended Chandra Deep Field South. By stacking the profiles of
187 LAEs at z = 2.06, 241 LAEs at z = 3.10, and 179 LAEs at z = 3.12, and
carefully performing low-surface brightness photometry, we obtain mean surface
brightness maps that reach 9.9, 8.7, and 6.2 * 10^{-19} ergs cm^{-2} s^{-1}
arcsec^{-2} in the emission line. We undertake a thorough investigation of
systematic uncertainties in our surface brightness measurements, and find that
our limits are 5--10 times larger than would be expected from Poisson
background fluctuations; these uncertainties are often underestimated in the
literature. At z ~ 3.1, we find evidence for extended halos with small scale
lengths of 5--8 kpc in some, but not all of our sub-samples. We demonstrate
that sub-samples of LAEs with low equivalent widths and brighter continuum
magnitudes are more likely to possess such halos. At z ~ 2.1, we find no
evidence of extended Ly-alpha emission down to our detection limits. Through
Monte-Carlo simulations, we also show that we would have detected large diffuse
LAE halos if they were present in our data sets. We compare these findings to
other measurements in the literature, and discuss possible instrumental and
astrophysical reasons for the discrepancies.
