Aims. Ly-alpha emitters (LAEs) can be detected out to very high redshifts during the epoch of reionization. The evolution of the LAE luminosity function with redshift is a direct probe of the Ly-alpha transmission of the intergalactic medium (IGM), and therefore of the IGM neutral-hydrogen fraction. Measuring the Ly-alpha luminosity function (LF) of LAEs at redshift z = 7.7 therefore allows us to constrain the ionizing state of the Universe at this redshift. Methods. We observed three 7.5x7.5 fields with the HAWK-I instrument at the VLT with a narrow band filter centred at 1.06 $mu$m and targeting LAEs at redshift z ~ 7.7. The fields were chosen for the availability of multiwavelength data. One field is a galaxy cluster, the Bullet Cluster, which allowed us to use gravitational amplification to probe luminosities that are fainter than in the field. The two other fields are subareas of the GOODS Chandra Deep Field South and CFHTLS-D4 deep field. We selected z=7.7 LAE candidates from a variety of colour criteria, in particular from the absence of detection in the optical bands. Results. We do not find any LAE candidates at z = 7.7 in ~2.4 x 10^4 Mpc^3 down to a narrow band AB magnitude of ~ 26, which allows us to infer robust constraints on the Ly-alpha LAE luminosity function at this redshift. Conclusions. The predicted mean number of objects at z = 6.5, derived from somewhat different LFs of Hu et al. (2010), Ouchi et al. (2010), and Kashikawa et al. (2011) are 2.5, 13.7, and 11.6, respectively. Depending on which of these LFs we refer to, we exclude a scenario with no evolution from z = 6.5 to z = 7.7 at 85% confidence without requiring a strong change in the IGM Ly-alpha transmission, or at 99% confidence with a significant quenching of the IGM Ly-alpha transmission, possibly from a strong increase in the high neutral-hydrogen fraction between these two redshifts.
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