We present a spectroscopic survey of Ly$alpha$ emitters (LAEs) at $zapprox3.1$ in the Subaru MM-Newton Deep Survey Field. This field has deep imaging data in a series of broad and narrow bands, including two adjacent narrow bands NB497 and NB503 that have allowed us to efficiently select LAE candidates at $zapprox3.1$. Using spectroscopic observations on MMT Hectospec and Magellan M2FS, we obtained a sample of 166 LAEs at $zapprox3.1$ over an effective area of $sim$1.2 deg$^2$, including 16 previously known LAEs. This is so far the largest (spectroscopically confirmed) sample of LAEs at this redshift. We make use of the secure redshifts and multi-band data to measure spectral properties such as Ly$alpha$ luminosity and rest-frame UV slope. We derive a robust Ly$alpha$ luminosity function (LF) that spans a luminosity range from $sim10^{42.0}$ to $>10^{43.5}$ erg s$^{-1}$. Significant overdense and underdense regions are detected in our sample, but the area coverage is wide enough to largely suppress the effect from such cosmic variance. Our Ly$alpha$ LF is generally consistent with those from previous studies at $z sim 3.1$. At the brightest end of the LF, there is a tentative detection of a density excess that is not well described by the Schechter function. The comparison with the LFs at other redshifts suggests that the Ly$alpha$ LF does not show significant evolution at $2<z<5$. Finally, we build the composite spectra of the LAEs and detect the NVI and CIV doublet emission lines at significance of $sim 4 sigma$, suggesting very hard radiation fields in (some of) these LAEs.
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