  We describe the results of a new, wide-field survey for z=3.1 Ly-alpha
emission-line galaxies (LAEs) in the Extended Chandra Deep Field South
(ECDF-S). By using a nearly top-hat 5010 Angstrom filter and complementary
broadband photometry from the MUSYC survey, we identify a complete sample of
141 objects with monochromatic fluxes brighter than 2.4E-17 ergs/cm^2/s and
observers-frame equivalent widths greater than ~ 80 Angstroms (i.e., 20
Angstroms in the rest-frame of Ly-alpha). The bright-end of this dataset is
dominated by x-ray sources and foreground objects with GALEX detections, but
when these interlopers are removed, we are still left with a sample of 130 LAE
candidates, 39 of which have spectroscopic confirmations. This sample overlaps
the set of objects found in an earlier ECDF-S survey, but due to our filter's
redder bandpass, it also includes 68 previously uncataloged sources. We confirm
earlier measurements of the z=3.1 LAE emission-line luminosity function, and
show that an apparent anti-correlation between equivalent width and continuum
brightness is likely due to the effect of correlated errors in our
heteroskedastic dataset. Finally, we compare the properties of z=3.1 LAEs to
LAEs found at z=2.1. We show that in the ~1 Gyr after z~3, the LAE luminosity
function evolved significantly, with L* fading by ~0.4 mag, the number density
of sources with L > 1.5E42 ergs/s declining by ~50%, and the equivalent width
scale-length contracting from 70^{+7}_{-5} Angstroms to 50^{+9}_{-6} Angstroms.
When combined with literature results, our observations demonstrate that over
the redshift range z~0 to z~4, LAEs contain less than ~10% of the
star-formation rate density of the universe.
