We analyze the volume-limited nearly complete 100 pc sample of 95 halo white dwarf candidates identified by the second data release of Gaia. Based on a detailed population synthesis model, we apply a method that relies on Gaia astrometry and photometry to accurately derive the individual white dwarf parameters (mass, radius, effective temperature, bolometric luminosity and age). This method is tested with 25 white dwarfs of our sample for which we took optical spectra and performed spectroscopic analysis. We build and analyse the halo white dwarf luminosity function, for which we find for the first time possible evidences of the cut-off at its faintest end, leading to an age estimate of $simeq12pm0.5 $Gyr. The mass distribution of the sample peaks at $0.589,M_{odot}$, with $71%$ of the white dwarf masses below $0.6,M_{odot}$ and just two massive white dwarfs of more than $0.8,M_{odot}$. From the age distribution we find three white dwarfs with total ages above 12 Gyr, of which J1312-4728 is the oldest white dwarf known with an age of $12.41pm0.22 $Gyr. We prove that the star formation history is mainly characterised by a burst of star formation that occurred from 10 to 12 Gyr in the past, but extended up to 8 Gyr. We also find that the peak of the star formation history is centered at around 11 Gyr, which is compatible with the current age of the Gaia-Enceladus encounter. Finally, $13%$ of our halo sample is contaminated by high-speed young objects (total age<7 Gyr). The origin of these white dwarfs is unclear but their age distribution may be compatible with the encounter with the Sagittarius galaxy.
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