We present a sample of 33 spectroscopically confirmed z ~ 3.1 Ly$alpha$-emitting galaxies (LAEs) in the Cosmological Evolution Survey (COSMOS) field. This paper details the narrow-band survey we conducted to detect the LAE sample, the optical spectroscopy we performed to confirm the nature of these LAEs, and a new near-infrared spectroscopic detection of the [O III] 5007 AA line in one of these LAEs. This detection is in addition to two [O III] detections in two z ~ 3.1 LAEs we have reported on previously (McLinden et al 2011). The bulk of the paper then presents detailed constraints on the physical characteristics of the entire LAE sample from spectral energy distribution (SED) fitting. These characteristics include mass, age, star-formation history, dust content, and metallicity. We also detail an approach to account for nebular emission lines in the SED fitting process - wherein our models predict the strength of the [O III] line in an LAE spectrum. We are able to study the success of this prediction because we can compare the model predictions to our actual near-infrared observations both in galaxies that have [O III] detections and those that yielded non-detections. We find a median stellar mass of 6.9 $times$ 10$^8$ M$_{odot}$ and a median star formation rate weighted stellar population age of 4.5 $times$ 10$^6$ yr. In addition to SED fitting, we quantify the velocity offset between the [O III] and Ly$alpha$ lines in the galaxy with the new [O III] detection, finding that the Ly$alpha$ line is shifted 52 km s$^{-1}$ redwards of the [O III] line, which defines the systemic velocity of the galaxy.
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