Context : Ultra-massive ( \gtrsim 1 M _ { \sun } ) hydrogen-rich ( DA ) white dwarfs are expected to have a substantial portion of their cores in a crystalline state at the effective temperatures characterizing the ZZ Ceti instability strip ( T _ { eff } \sim 12 500 K ) , as a result of Coulomb interactions in very dense plasmas .
Asteroseismological analyses of these white dwarfs can provide valuable information related to the crystallization process , the core chemical composition and the evolutionary origin of these stars .

Aims : 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 white dwarf 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 .

Methods : We considered g -mode adiabatic pulsation periods for ultra-massive ONe-core DA white dwarf models with stellar masses in the range 1.10 \lesssim M _ { \star } / M _ { \sun } \lesssim 1.29 , effective temperatures in the range 10 000 \lesssim T _ { eff } \lesssim 15 000 K , and H envelope thicknesses in the interval -10 \lesssim \log ( M _ { H } / M _ { \star } ) \lesssim - 6 .
We explore the effects of employing different H-envelope thicknesses on the mode-trapping properties of our ultra-massive ONe-core DA white dwarf models , and perform period-to-period fits to ultra-massive ZZ Ceti stars with the aim of finding an asteroseismological model for each target star .

Results : We found that the trapping cycle and trapping amplitude are larger for thinner H envelopes , and that the asymptotic period spacing is longer for thinner H envelopes .
We found a mean period spacing of \Delta \Pi \sim 17 s in the data of BPM 37093 , which is likely to be associated to \ell = 2 modes .
However , we are not able to put constraints on the stellar mass of BPM 37093 using this mean period spacing due to the simultaneous sensitivity of \Delta \Pi with M _ { \star } , T _ { eff } , and M _ { H } , an intrinsic property of DAV stars .
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 .
The effective temperature and stellar mass of these models are in agreement with the spectroscopic determinations .
The percentage of crystallized mass of these asteroseismological models is 92 % , 97 % , and 81 % for BPM 37093 , GD 518 , and SDSS J0840+5222 , respectively .
We also derive asteroseismological distances which are in agreement with the astrometric measurements of Gaia for these stars .

Conclusions : Asteroseismological analyses like the one presented in this paper could lead to a more complete knowledge of the processes occurring during crystallization inside white dwarfs .
Also , they could make it possible to deduce the core chemical composition of ultra-massive white dwarfs , 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 .
However , to achieve these objectives it is necessary to find more pulsating ultra-massive WDs , and also to carry out additional observations of the already known pulsating stars to detect more pulsation periods .
Space missions such as TESS can give a great boost in this direction .