  We searched for quasars that are ~ 3 mag fainter than the SDSS quasars in the
redshift range 3.7 < z < 4.7 in the COSMOS field to constrain the faint end of
the quasar luminosity function. Using optical photometric data, we selected 31
quasar candidates with 22 < i' < 24 at z ~ 4. We obtained optical spectra for
most of these candidates using FOCAS on the Subaru telescope, and identified 8
low-luminosity quasars at z ~ 4. In order to derive the quasar luminosity
function (QLF) based on our spectroscopic follow-up campaign, we estimated the
photometric completeness of our quasar survey through detailed Monte Carlo
simulations. Our QLF at z ~ 4 has a much shallower faint-end slope beta =
-1.67^{+0.11}_{-0.17} than that obtained by other recent surveys in the same
redshift. Our result is consistent with the scenario of downsizing evolution of
active galactic nuclei inferred by recent optical and X-ray quasar surveys at
lower redshifts.
