  We study the global stability of non-axisymmetric p-modes (also called
inertial-acoustic modes) trapped in the inner-most regions of accretion discs
around black holes. We show that the lowest-order (highest-frequency) p-modes,
with frequencies $\omega=(0.5-0.7) m\Omega_{\rm ISCO}$, can be overstable due
to general relativistic effects, according to which the radial epicyclic
frequency is a non-monotonic function of radius near the black hole. The mode
is trapped inside the corotation resonance radius and carries a negative
energy. The mode growth arises primarily from wave absorption at the corotation
resonance, and the sign of the wave absorption depends on the gradient of the
disc vortensity. When the mode frequency is sufficiently high, such that the
slope of the vortensity is positive at corotation positive wave energy is
absorbed at the resonance, leading to the growth of mode amplitude. We also
study how the rapid radial inflow at the inner edge of the disc affects the
mode trapping and growth. Our analysis of the behavior of the fluid
perturbations in the transonic flow near the ISCO indicates that, while the
inflow tends to damp the mode, the damping effect is sufficiently small under
some conditions so that net mode growth can still be achieved. We further
clarify the role of the Rossby wave instability and show that it does not
operate for black hole accretion discs with smooth-varying vortensity profiles.
Overstable non-axisymmetric p-modes driven by the corotational instability
provide a plausible explanation for the high-frequency (> 100 Hz)
quasi-periodic oscillations (HFQPOs) observed from a number of black-hole X-ray
binaries in the very high state. The absence of HFQPOs in the soft (thermal)
state may result from mode damping due to the radial infall at the ISCO.
