Gaia Early Data Release 3: The Gaia Catalogue of Nearby Stars

We produce a clean and well-characterised catalogue of objects within 100\,pc of the Sun from the \G\ Early Data Release 3. We characterise the catalogue through comparisons to the full data release, external catalogues, and simulations. We carry out a first analysis of the science that is possible with this sample to demonstrate its potential and best practices for its use. The selection of objects within 100\,pc from the full catalogue used selected training sets, machine-learning procedures, astrometric quantities, and solution quality indicators to determine a probability that the astrometric solution is reliable. The training set construction exploited the astrometric data, quality flags, and external photometry. For all candidates we calculated distance posterior probability densities using Bayesian procedures and mock catalogues to define priors. Any object with reliable astrometry and a non-zero probability of being within 100\,pc is included in the catalogue. We have produced a catalogue of \NFINAL\ objects that we estimate contains at least 92\% of stars of stellar type M9 within 100\,pc of the Sun. We estimate that 9\% of the stars in this catalogue probably lie outside 100\,pc, but when the distance probability function is used, a correct treatment of this contamination is possible. We produced luminosity functions with a high signal-to-noise ratio for the main-sequence stars, giants, and white dwarfs. We examined in detail the Hyades cluster, the white dwarf population, and wide-binary systems and produced candidate lists for all three samples. We detected local manifestations of several streams, superclusters, and halo objects, in which we identified 12 members of \G\ Enceladus. We present the first direct parallaxes of five objects in multiple systems within 10\,pc of the Sun.