Abridged: A photometric sample of ~7100 V<25.3 Lyman break galaxies (LBGs) has been selected by combining Subaru/Suprime-Cam BVRciz data with deep GALEX/NUV imaging of the Subaru Deep Field. Follow-up spectroscopy confirmed 24 LBGs at 1.5<z<2.7. Among the optical spectra, 12 have Ly-alpha emission with rest-frame equivalent widths of ~5-60AA. The success rate for identifying LBGs as NUV-dropouts at 1.5<z<2.7 is 86%. The rest-frame UV (1700AA) luminosity function (LF) is constructed from the photometric sample with corrections for stellar contamination and z<1.5 interlopers. The LF is 1.7+/-0.1 times higher than those of z~2 BXs and z~3 LBGs. Three explanations were considered, and it is argued that significantly underestimating low-z contamination or effective comoving volume is unlikely: the former would be inconsistent with the spectroscopic sample at 93% confidence, and the second explanation would not resolve the discrepancy. The third scenario is that different photometric selection of the samples yields non-identical galaxy populations, such that some BX galaxies are LBGs and vice versa. This argument is supported by a higher surface density of LBGs at all magnitudes while the redshift distribution of the two populations is nearly identical. This study, when combined with other star-formation rate (SFR) density UV measurements from LBG surveys, indicates that there is a rise in the SFR density: a factor of 3-6 (3-10) increase from z~5 (z~6) to z~2, followed by a decrease to z~0. This result, along with past sub-mm studies that find a peak at z~2 in their redshift distribution, suggest that z~2 is the epoch of peak star-formation. Additional spectroscopy is required to characterize the complete shape of the z~2 LBG UV LF via measurements of contamination and accurate distances.
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