Context :

Aims : We perform a deep search for galaxies in the redshift range 6.5 \leq z \leq 7.5 , to measure the evolution of the number density of luminous galaxies in this redshift range and derive useful constraints on the evolution of their luminosity function .

Methods : We present here the second half of an ESO Large Programme , which exploits the unique combination of area and sensitivity provided in the near–IR by the camera Hawk-I at the VLT .
We have obtained \sim 30 observing hours with Hawk-I in the Y -band of two high galactic latitude fields .
We combined the Y -band data with deep J and K Hawk-I observations , and with FORS1/FORS2 U , B , V , R , I , and Z observations to select z -drop galaxies having Z - Y > 1 , no optical detection and flat Y - J and Y - K colour terms .

Results : We detect 8 high-quality candidates in the magnitude range Y = 25.5 - 26.5 that we add to the z -drop candidates selected in two Hawk-I pointings over the GOODS-South field .
We use this full sample of 15 objects found in \sim 161 ~ { } arcmin ^ { 2 } of our survey to constrain the average physical properties and the evolution of the number density of z \sim 7 LBGs .
A stacking analysis yields a best-fit SED with photometric redshift z = 6.85 ^ { +0.20 } _ { -0.15 } and an E ( B - V ) = 0.05 ^ { +0.15 } _ { -0.05 } .
We compute a binned estimate of the z \sim 7 LF and explore the effects of photometric scatter and model uncertainties on the statistical constraints .
After accounting for the expected incompleteness through MonteCarlo simulations , we strengthen our previous finding that a Schechter luminosity function constant from z=6 to z=7 is ruled out at a \gtrsim 99 % confidence level , even including the effects of cosmic variance .
For galaxies brighter than M _ { 1500 } = -19.0 , we derive a luminosity density \rho _ { UV } = 1.5 ^ { +2.1 } _ { -0.8 } \times 10 ^ { 25 } erg~ { } s ^ { -1 } ~ { } Hz ^ { -1 } ~ { } Mpc ^ { -3 } , implying a decrease by a factor 3.5 from z = 6 to z \simeq 6.8 .
We find that , under standard assumptions , the emission rate of ionizing photons coming from UV bright galaxies is lower by at least a factor of two than the value required for reionization .
Finally , we exploit deep Hawk-I J and K band observations to derive an upper limit on the number density of M _ { 1500 } \lesssim - 22.0 LBGs at z \sim 8 ( Y -dropouts ) .

Conclusions :