  We study global non-axisymmetric oscillation modes and instabilities in
magnetosphere- disc systems, as expected in neutron star X-ray binaries and
possibly also in accreting black hole systems. Our two-dimensional
magnetosphere-disc model consists of a Keplerian disc in contact with an
uniformly rotating magnetosphere with low plasma density. Two types of global
overstable modes exist in such systems, the interface modes and the disc
inertial-acoustic modes. We examine various physical effects and parameters
that influence the properties of these oscillation modes, particularly their
growth rates, including the magnetosphere field configuration, the velocity and
density contrasts across the magnetosphere-disc interface, the rotation profile
(with Newtonian or General Relativistic potential), the sound speed and
magnetic field of the disc. The interface modes are driven unstable by
Rayleigh-Taylor and Kelvin-Helmholtz in- stabilities, but can be stabilized by
the toroidal field (through magnetic tension) and disc differential rotation
(through finite vorticity). General relativity increases their growth rates by
modifying the disc vorticity outside the magnetosphere boundary. The interface
modes may also be affected by wave absorption associated with corotation
resonance in the disc. In the presence of a magnetosphere, the
inertial-acoustic modes are effectively trapped at the innermost region of the
relativistic disc just outside the interface. They are driven unstable by wave
absorption at the corotation resonance, but can be stabilized by modest disc
magnetic fields. The overstable oscillation modes studied in this paper have
characteristic properties that make them possible candidates for the
quasi-periodic oscillations observed in X-ray binaries.
