  Ultra-massive hydrogen-rich (DA spectral type) white dwarf (WD) stars
($M_{\star} > 1M_{\odot}$) coming from single-star evolution are expected to
harbor cores made of $^{16}$O and $^{20}$Ne, resulting from semi-degenerate
carbon burning when the progenitor star evolves through the super asymptotic
giant branch (S-AGB) phase. These stars are expected to be crystallized by the
time they reach the ZZ Ceti instability strip ($T_{\rm eff} \sim 12\,500$ K).
Theoretical models predict that crystallization leads to a separation of
$^{16}$O and $^{20}$Ne in the core of ultra-massive WDs, which impacts their
pulsational properties. This property offers a unique opportunity to study the
processes of crystallization. Here, we present the first results of a detailed
asteroseismic analysis of the best-studied ultra-massive ZZ Ceti star
BPM~37093. As a second step, we plan to repeat this analysis using
ultra-massive DA WD models with C/O cores in order to study the possibility of
elucidating the core chemical composition of BPM~37093 and shed some light on
its possible evolutionary origin. We also plan to extend this kind of analyses
to other stars observed from the ground and also from space missions like
Kepler and TESS.
