  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 7.5'x7.5' 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.
