  We present a spectroscopic survey of Ly$\alpha$ emitters (LAEs) at $z\sim3.7$
and $z\sim4.8$. The LAEs are selected using the narrowband technique based on
the combination of deep narrowband and broadband imaging data in two deep
fields, and then spectroscopically confirmed with the MMT multi-fiber
spectrograph Hectospec. The sample consists of 71 LAEs at $z\sim3.7$ and 69
LAEs at $z\sim4.8$ over $\sim 1.5$ deg$^2$, making it one of the largest
spectroscopically confirmed sample of LAEs at the two redshifts. Their
Ly$\alpha$ luminosities are measured using the secure redshifts and deep
photometric data, and span a range of $\sim 10^{42.5}$ - $10^{43.6} \,\rm erg\,
s^{-1}$, so these LAEs represent the most luminous galaxies at the redshifts in
terms of Ly$\alpha$ luminosity. We estimate and correct sample incompletenesses
and derive reliable Ly$\alpha$ luminosity function (LF)s at $z\sim3.7$ and 4.8
based on the two spectroscopic samples. We find that our Ly$\alpha$ LFs are
roughly consistent (within a factor of $2-3$) with previous measurements at
similar redshifts that were derived from either photometric samples or
spectroscopic samples. By comparing with previous studies in different
redshifts, we find that the Ly$\alpha$ LFs decrease mildly from $z\sim3.1$ to
$z\sim5.7$, supporting the previous claim of the slow LF evolution between
$z\sim2$ and $z\sim6$. At $z>5.7$, the LF declines rapidly towards higher
redshift, partly due to the effect of cosmic reionization.
