  Context: Around 30 per cent of the observed exoplanets that orbit M dwarf
stars are gas giants that are more massive than Jupiter. These planets are
prime candidates for formation by disc instability. Aims: We want to determine
the conditions for disc fragmentation around M dwarfs and the properties of the
planets that are formed by disc instability. Methods: We performed hydrodynamic
simulations of M dwarf protostellar discs in order to determine the minimum
disc mass required for gravitational fragmentation to occur. Different stellar
masses, disc radii, and metallicities were considered. The mass of each
protostellar disc was steadily increased until the disc fragmented and a
protoplanet was formed. Results: We find that a disc-to-star mass ratio between
$\sim 0.3$ and $\sim 0.6$ is required for fragmentation to happen. The minimum
mass at which a disc fragments increases with the stellar mass and the disc
size. Metallicity does not significantly affect the minimum disc fragmentation
mass but high metallicity may suppress fragmentation. Protoplanets form quickly
(within a few thousand years) at distances around $\sim50$ AU from the host
star, and they are initially very hot; their centres have temperatures similar
to the ones expected at the accretion shocks around planets formed by core
accretion (up to 12,000K). The final properties of these planets (e.g. mass and
orbital radius) are determined through long-term disc-planet or planet-planet
interactions. Conclusions: Disc instability is a plausible way to form gas
giant planets around M dwarfs provided that discs have at least 30% the mass of
their host stars during the initial stages of their formation. Future
observations of massive M dwarf discs or planets around very young M dwarfs are
required to establish the importance of disc instability for planet formation
around low-mass stars.
