We investigate the high-redshift quasar luminosity function (QLF) down to an apparent magnitude of I(AB) = 25 in the Cosmic Evolution Survey (COSMOS). Careful analysis of the extensive COSMOS photometry and imaging data allows us to identify and remove stellar and low-redshift contaminants, enabling a selection that is nearly complete for type-1 quasars at the redshifts of interest. We find 155 likely quasars at z > 3.1, 39 of which have prior spectroscopic confirmation. We present our sample in detail and use these confirmed and likely quasars to compute the rest-frame UV QLF in the redshift bins 3.1 < z < 3.5 and 3.5 < z < 5. The space density of faint quasars decreases by roughly a factor of four from z sim 3.2 to z sim 4, with faint-end slopes of {beta} sim -1.7 at both redshifts. The decline in space density of faint optical quasars at z > 3 is similar to what has been found for more luminous optical and X-ray quasars. We compare the rest-frame UV luminosity functions found here with the X-ray luminosity function at z > 3, and find that they evolve similarly between z sim 3.2 and z sim 4; however, the different normalizations imply that roughly 75% of X-ray bright active galactic nuclei (AGN) at z sim 3 - 4 are optically obscured. This fraction is higher than found at lower redshift and may imply that the obscured, type-2 fraction continues to increase with redshift at least to z sim 4. Finally, the implications of the results derived here for the contribution of quasars to cosmic reionization are discussed.
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