  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.
