We searched for quasars that are \sim 3 mag fainter than the SDSS quasars in the redshift range 3.7 \lesssim z \lesssim 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 \textless i \arcmin \textless 24 at z \sim  4 .
We obtained optical spectra for most of these candidates using FOCAS on the Subaru telescope , and identified 8 low-luminosity quasars at z \sim 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 \sim 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 .