  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 $\simeq12\pm0.5 $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 $12.41\pm0.22 $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.
