  Ultra-massive white dwarfs are relevant for their role as type Ia Supernova
progenitors, the occurrence of physical processes in the asymptotic
giant-branch phase, the existence of high-field magnetic white dwarfs, and the
occurrence of double white dwarf mergers. Some hydrogen-rich ultra-massive
white dwarfs are pulsating stars, and as such, they offer the possibility of
studying their interiors through asteroseismology. On the other hand, pulsating
helium-rich ultra-massive white dwarfs could be even more attractive objects
for asteroseismology if they were found, as they should be hotter and less
crystallized than pulsating hydrogen-rich white dwarfs, something that would
pave the way for probing their deep interiors. We explore the pulsational
properties of ultra-massive helium-rich white dwarfs with carbon-oxygen and
oxygen-neon cores resulting from single stellar evolution. Our goal is to
provide a theoretical basis that could eventually help to discern the core
composition of ultra-massive white dwarfs and the scenario of their formation
through asteroseismology, anticipating the possible future detection of
pulsations in this type of stars. We find that, given that the white dwarf
models coming from the three scenarios considered are characterized by distinct
core chemical profiles, their pulsation properties are also different, thus
leading to distinctive signatures in the period-spacing and mode-trapping
properties. Our results indicate that, in case of an eventual detection of
pulsating ultra-massive helium-rich white dwarfs, it would be possible to
derive valuable information encrypted in the core of these stars in connection
with the origin of such exotic objects. The detection of pulsations in these
stars has many chances to be achieved soon through observations collected with
ongoing space missions.
