We use the large homogeneous sample of late M dwarfs, M7 to M9.5, of Ahmed & Warren (2019) matched to Gaia DR2, to measure the relation between absolute magnitude and spectral type, and to infer the multiplicity fraction of the population, and the distribution of mass ratios in the binary systems. Binaries are identified photometrically as overluminous sources. In order to define a sample that is unbiased with respect to multiplicity we use distance limits that are a function of G − J colour to define a volume-complete sample of 2706 systems. The G − J colours are very precise, with random errors all less than 0.02. We measure absolute magnitudes MJ that are on average 0.5 mag. brighter than previous determinations. We find evidence that the discrepancies arise from differences in spectral types in different samples. The measured binary fraction is $16.5\pm 0.8\%$, of which $98\%$ are unresolved: both values are consistent with results of previous studies. The distribution of excess flux in the binaries, compared to the singles, is used to infer the mass ratio distribution f(q)∝qγ, where q = Ms/Mp. We infer a very steep distribution over this spectral range, with γ > 10 ($99\%$ probability). This says that unresolved ultracool M dwarf binaries reside almost exclusively in equal mass systems, and implies that the spectral types of the unresolved binaries match to with 0.5 spectral subtypes. The intrinsic scatter in absolute magnitude MJ for ultracool M dwarfs at fixed G − J colour is measured to be 0.21 mag.

We investigate the form of the local vertical density profile of the stars in the Galactic disk, close to the Galactic plane. We use a homogeneous sample of 34 000 ultracool dwarfs M7 to L2.5 that all lie within 350 pc of the plane. We fit a profile of the form sech α , where α = 2 is the theoretically preferred isothermal profile and α = 0 is the exponential function. Larger values of α correspond to greater flattening of the profile towards the plane. We employ a likelihood analysis that accounts in a direct way for unresolved binaries in the sample, as well as for the spread in absolute magnitude M J within each spectral sub-type (Malmquist bias). We measure α = 0.29 +0.12 −0.13 . The α = 1 (sech) and flatter profiles are ruled out at high confidence for this sample, while α = 0 (exponential) is included in the 95% credible interval. Any flattening relative to exponential is modest, and is confined to within 50 pc of the plane. The measured value of α is consistent with the results of the recent analysis by Xiang et al. Our value for α is also similar to that determined for nearby spiral galaxies by de Grijs et al., measured from photometry of galaxies viewed edge on. The measured profile allows an accurate determination of the local space density of ultracool dwarfs M7 to L2.5, and we use this to make a new determination of the luminosity function at the bottom of the main sequence. Our results for the luminosity function are a factor two to three lower than the recent measurement by Bardalez Gagliuffi et al., that uses stars in the local 25 pc radius bubble, but agree well with the older study by Cruz et al.