  State-of-the-art models of massive black hole formation postulate that
quasars at $z>6$ reside in extreme peaks of the cosmic density structure in the
early universe. Even so, direct observational evidence of these overdensities
is elusive, especially on large scales ($\gg$1 Mpc) as the spectroscopic
follow-up of $z>6$ galaxies is observationally expensive. Here we present Keck
/ DEIMOS optical and IRAM / NOEMA millimeter spectroscopy of a $z\sim6$
Lyman-break galaxy candidate originally discovered via broadband selection, at
a projected separation of 4.65 physical Mpc (13.94 arcmin) from the luminous
$z$=6.308 quasar J1030+0524. This well-studied field presents the strongest
indication to date of a large-scale overdensity around a $z>6$ quasar. The Keck
observations suggest a $z\sim6.3$ dropout identification of the galaxy. The
NOEMA 1.2mm spectrum shows a 3.5$\sigma$ line that, if interpreted as [CII],
would place the galaxy at $z$=6.318 (i.e., at a line-of-sight separation of 3.9
comoving Mpc assuming that relative proper motion is negligible). The measured
[CII] luminosity is $3\times10^8$ L$_\odot$, in line with expectations for a
galaxy with a star formation rate $\sim15$ M$_\odot$ yr$^{-1}$, as inferred
from the rest-frame UV photometry. Our combined observations place the galaxy
at the same redshift as the quasar, thus strengthening the overdensity scenario
for this $z>6$ quasar. This pilot experiment demonstrates the power of
millimeter-wavelength observations in the characterization of the environment
of early quasars.
