  We prove the existence of instabilities for the geometric linear wave
equation on extremal Kerr spacetime backgrounds, which describe stationary
black holes rotating at their maximally allowed angular velocity. These
instabilities can be associated to non-axisymmetric azimuthal modes and are
stronger than the axisymmetric instabilities discovered by Aretakis in [Are15].
The existence of non-axisymmetric instabilities follows from a derivation of
very precise stability properties of solutions: we determine therefore the
precise, global, leading-order, late-time behaviour of solutions supported on a
bounded set of azimuthal modes via energy estimates in both physical and
frequency space. In particular, we obtain sharp, uniform decay-in-time
estimates and we determine the coefficients and rates of inverse-polynomial
late-time tails everywhere in the exterior of extremal Kerr black holes. We
also demonstrate how non-axisymmetric instabilities leave an imprint on future
null infinity via the coefficients appearing in front of slowly decaying and
oscillating late-time tails.
