  We present luminosity functions (LFs) and various properties of Lya emitters
(LAEs) at z=3.1, 3.7, and 5.7, in a 1 deg^2 sky of the Subaru/XMM-Newton Deep
Survey (SXDS) Field. We obtain a photometric sample of 858 LAE candidates based
on deep Subaru/Suprime-Cam imaging data, and a spectroscopic sample of 84
confirmed LAEs from Subaru/FOCAS and VLT/VIMOS spectroscopy in a survey volume
of ~10^6 Mpc^3 with a limiting Lya luminosity of ~3x10^42 erg/s. We derive the
LFs of Lya and UV-continuum (~1500 \AA) for each redshift, taking into account
the statistical error and the field-to-field variation. We find that the
apparent Lya LF shows no significant evolution between z=3.1 and 5.7 within
factors of 1.8 and 2.7 in L* and phi*, respectively. On the other hand, the UV
LF of LAEs increases from z=3.1 to 5.7, indicating that galaxies with Lya
emission are more common at earlier epochs. We identify six LAEs with AGN
activities from our spectra combined with VLA, Spitzer, and XMM-Newton data.
Among the photometrically selected LAEs at z=3.1 and 3.7, only ~1 % show AGN
activities, while the brightest LAEs with logL(Lya) >~ 43.4-43.6 erg/s appear
to always host AGNs. Our LAEs are bluer in UV-continuum color than dropout
galaxies, suggesting lower extinction and/or younger stellar populations. Our
stacking analyses provide upper limits to the radio luminosity and the
f(HeII)/f(Lya) line fraction, and constrain the hidden star formation
(+low-luminosity AGN) and the primordial population in LAEs.
