A Spectroscopic Survey of Ly$\alpha$ Emitters at $z\approx3.1$ over
  $\sim$1.2 Deg$^2$
  We present a spectroscopic survey of Ly$\alpha$ emitters (LAEs) at
$z\approx3.1$ 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 $z\approx3.1$. Using spectroscopic observations on MMT Hectospec
and Magellan M2FS, we obtained a sample of 166 LAEs at $z\approx3.1$ 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.
