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.
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