  Low-order, non-axisymmetric p-modes (also referred as inertial-acoustic
modes) trapped in the inner-most region of hydrodynamic accretion discs around
black holes, are plausible candidates for high-frequency quasi-periodic
oscillations (QPOs) observed in a number of accreting black-hole systems. These
modes are subject to global instabilities due to wave absorption at the
corotation resonance (where the wave pattern frequency $\omega/m$ equals the
disc rotation rate $\Omega$), when the fluid vortensity,
$\zeta=\kappa^2/(2\Omega\Sigma)$ (where $\kappa$ and $\Sigma$ are the radial
epicyclic frequency and disc surface density, respectively), has a positive
gradient. We investigate the effects of disc magnetic fields on the wave
absorption at corotation and the related wave super-reflection of the
corotation barrier, and on the overstability of disc p-modes. For discs with a
pure toroidal field, the corotation resonance is split into two magnetic
resonances, where the wave frequency in the corotating frame of the fluid,
$\tomega=\omega-m\Omega$, matches the slow magnetosonic wave frequency.
Significant wave energy/angular momentum absorption occurs at both magnetic
resonances, but with opposite signs. The combined effect of the two magnetic
resonances is to reduce the super-reflection and the growth rate of the
overstable p-modes. We show that even a subthermal toroidal field may suppress
the overstability of hydrodynamic (B=0) p-modes. For accretion discs with mixed
(toroidal and vertical) magnetic fields, two additional Alfven resonances
appear, where $\tomega$ matches the local Alfven wave frequency. They further
reduce the growth rate of p-modes. Our results suggest that in order for the
non-axisymmetric p-modes to be a viable candidate for the observed
high-frequency QPOs, the disc magnetic field must be appreciably subthermal, or
other mode excitation mechanisms are at work.
