  We present additional optical spectroscopic identifications of sources
identified in three deep ASCA GIS fields which also form part of a deep ROSAT
survey. In total, 26 ASCA sources have been detected down to a 2-10keV flux
limit of S = 5e-14 erg/cm/s. LDSS observations have increased the spectroscopic
completeness of the survey to between 65 and 85 per cent, with identifications
for up to 13 QSOs with broad emission lines and 6 objects with narrow emission
lines. Combining these objects with the AGN identified in the LMA/HEAO-1 sample
by Grossan, we find evidence for significant cosmological evolution in the
2-10keV band (<V_e/V_a>=0.63+/-0.03), fit by a pure luminosity evolution model;
L \propto (1+z)^k, k=2.04^{+0.16}_{-0.22}$. The present-epoch 2-10keV AGN X-ray
luminosity function, \Phi(L_X), is best represented by a two power law
function: \Phi(L_X) \propto L^{-3.0}, L*>10^{44.1}erg/s; $\Phi (L_X) \propto
L^{-1.9}$, L*<10^{44.1}erg/s. Depending on the extent to which we extrapolate
the z=0 AGN luminosity function, we predict a total contribution to the 2-10keV
X-ray background from QSOs and narrow-emission-line galaxies which ranges from
48 per cent (for AGN with L>10^{42}erg/s) to 80 per cent (L>10^{39}erg/s).
