  In a previous work, we showed that the differentially rotating equatorial
disk in the pulsar's electrosphere is diocotron unstable in the
non-relativistic regime. In this paper, we extend these results and study the
relativistic and electromagnetic stabilisation effects by including the
magnetic field perturbation and allow for relativistic speeds of the guiding
centre, in a self-consistent manner. We use the electric drift approximation,
valid for low-density plasmas.
  We linearise the coupled relativistic cold-fluid and Maxwell equations in the
electric drift approximation. The non-linear eigenvalue problem for the
perturbed azimuthal electric field is solved numerically with standard technics
for boundary value problems like the shooting method. The spectrum of the
relativistic diocotron instability in a non-neutral plasma column confined
between two cylindrically conducting walls is computed.
  For low-speed motions, we recover the eigenfunctions and eigenspectra of the
non-relativistic diocotron instability. Our algorithm is also checked in the
relativistic planar diode geometry for which an analytical expression of the
dispersion relation is known. As expected, when the relativistic and
electromagnetic effects become significant, the diocotron instability tends to
stabilise. In cylindrical geometry, for some special rotation profile, all
azimuthal modes $l$ are completely suppressed for sufficiently relativistic
flows. However, for the profile relevant to the electrosphere, depending on the
exact rotation curves, the growth rates can either significantly decrease till
they vanish or persist for moderate $l$.
