  Ultra-massive hydrogen-rich (DA) white dwarfs (WD) are expected to have a
substantial portion of their cores in a crystalline state at the effective
temperatures characterizing the ZZ Ceti instability strip, as a result of
Coulomb interactions. Asteroseismological analyses of these WDs can provide
valuable information related to the crystallization process, the core chemical
composition and the evolutionary origin of these stars. We present a thorough
asteroseismological analysis of the ultra-massive ZZ Ceti star BPM~37093, which
exhibits a rich period spectrum, on the basis of a complete set of fully
evolutionary models that represent ultra-massive oxygen/neon(ONe)-core DA WD
stars harbouring a range of hydrogen (H) envelope thicknesses. We also carry
out preliminary asteroseismological inferences on two other ultra-massive ZZ
Ceti stars that exhibit fewer periods, GD~518, and SDSS~J0840+5222. We found a
mean period spacing of $\Delta \Pi \sim 17$ s in the data of BPM~37093,
associated to $\ell= 2$ modes. We found asteroseismological models for the
three objects under analysis, two of them (BPM~37093 and GD~518) characterized
by canonical (thick) H envelopes, and the third one (SDSS~J0840+5222) with a
thinner H envelope. We also derive asteroseismological distances which are
somewhat different to the astrometric measurements of {\it Gaia} for these
stars. Asteroseismological analyses like the one presented in this paper could
lead to a more complete knowledge of the processes occurring during
crystallization inside WDs. Also, they could make it possible to deduce the
core chemical composition of ultra-massive WDs, and in this way, to infer their
evolutionary origin, i.e., if the star has a ONe core and then is the result of
single-star evolution, or if it harbour a carbon/oxygen (CO) core and is the
product of a merger of the two components of a binary system.
