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

Evolution of the observed Ly-alpha luminosity function from z = 6.5 to z = 7.7: evidence for the epoch of reionization ?