We consider the axisymmetric global modes of a uniformly magnetized ac
cretion disc. We use a local disc model that incorporates vertical str
atification but, except for velocity shear, ignores radial structure.
Analytic expressions are obtained for the modes and their growth rates
in the special case of a uniform vertical field, and the modes and gr
owth rates are obtained numerically for more general orientations of t
he magnetic field. We find that, in general, the growing modes are ove
rstable. We also report a new, global version of the instability of Ba
lbus & Hawley that sets in for some, but not all, boundary conditions.
This global instability can occur even when the magnetic field is str
ong. The existence of this global instability explains the difference
between the instability criterion of Papaloizou & Szuszkiewicz and the
instability criterion derived from a local analysis by Balbus & Hawle
y.