We studied the clustering properties and multiwavelength spectral energy distributions of a complete sample of 162 Ly Alpha-Emitting (LAE) galaxies at z=3.1 discovered in deep narrow-band MUSYC imaging of the Extended Chandra Deep Field South. LAEs were selected to have observed frame equivalent widths >80A and emission line fluxes >1.5E-17 erg/cm^2/s. Only 1% of our LAE sample appears to host AGN. The LAEs exhibit a moderate spatial correlation length of r_0=3.6+0.8-1.0 Mpc, corresponding to a bias factor b=1.7+0.3-0.4, which implies median dark matter halo masses of log10(M_med) = 10.9+0.5-0.9 M_sun. Comparing the number density of LAEs, (1.5+-0.3)E-3/Mpc^3, with the number density of these halos finds a mean halo occupation ~1-10%. The evolution of galaxy bias with redshift implies that most z=3.1 LAEs evolve into present-day galaxies with L<2.5L*, whereas other z>3 galaxy populations typically evolve into more massive galaxies. Halo merger trees show that z=0 descendants occupy halos with a wide range of masses, with a median descendant mass close to that of L*. Only 30% of LAEs have sufficient stellar mass (>~3E9 M_sun) to yield detections in deep Spitzer-IRAC imaging. A two-population SED fit to the stacked UBVRIzJK+[3.6,4.5,5.6,8.0]micron fluxes of the IRAC-undetected objects finds that the typical LAE has low stellar mass (1.0+0.6-0.4 E9 M_sun), moderate star formation rate (2+-1 M_sun/yr), a young component age of 20+30-10 Myr, and little dust (A_V<0.2). The best fit model has 20% of the mass in the young stellar component, but models without evolved stars are also allowed.
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