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 \sim 50 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 12,000 K ) .
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 .