  Aims: In this work the luminosity function of QSOs is measured in the
redshift range 3.5<z<5.2 for the absolute magnitude interval -21<M_{145}<-28.
The determination of the faint end of the luminosity function at these
redshifts provides important constraints on models of joint evolution of
galaxies and AGNs. Methods: We have defined suitable criteria to select faint
QSOs in the GOODS fields, checking in detail their effectiveness and
completeness. Spectroscopic follow-up of the resulting QSO candidates has been
carried out. The confirmed sample of faint QSOs is compared with a brighter one
derived from the SDSS. We have used a Monte-Carlo technique to estimate the
properties of the luminosity function, checking various parameterizations for
its shape and evolution. Results: Models based on Pure Density Evolution show
better agreement with observation than models based on Pure Luminosity
Evolution. However a different break magnitude with respect to z~2.1 is
required at 3.5<z<5.2. Models with a steeper faint end score a higher
probability. We do not find any evidence for a flattening of the bright end at
redshift z>3.5. Conclusions: The estimated space density evolution of QSOs
indicates a suppression of the formation and/or feeding of Supermassive Black
Holes at these redshifts. The QSO contribution to the UV background is
insufficient to ionize the IGM at 3.5<z<5.2.
