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 $zsim6$ 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 $zsim6.3$ 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 $3times10^8$ 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.
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