  Recent evidence suggests that high-redshift Ly-alpha emitting galaxies (LAEs)
with logL(Ly-alpha) > 43.5 erg/s, referred to as ultraluminous LAEs (ULLAEs),
may show less evolution than lower-luminosity LAEs in the redshift range
z=5.7-6.6. Here we explore the redshift evolution of the velocity widths of the
Ly-alpha emission lines in LAEs over this redshift interval. We use new
wide-field, narrowband observations from Subaru/Hyper Suprime-Cam to provide a
sample of 24 z=6.6 and 12 z=5.7 LAEs with log L(Ly-alpha) > 43 erg/s, all of
which have follow-up spectroscopy from Keck/DEIMOS. Combining with archival
lower-luminosity data, we find a significant narrowing of the Ly-alpha lines in
LAEs at logL(Ly-alpha) < 43.25 erg/s -- somewhat lower than the usual ULLAE
definition -- at z = 6.6 relative to those at z = 5.7, but we do not see this
in higher-luminosity LAEs. As we move to higher redshifts, the increasing
neutrality of the intergalactic medium should increase the scattering of the
Ly-alpha lines, making them narrower. The absence of this effect in the
higher-luminosity LAEs suggests they may lie in more highly ionized regions,
self-shielding from the scattering effects of the intergalactic medium.

Astrophysics of Galaxies