  In this paper we introduce a notion of magnetic field in the Heisenberg group
and we study its influence on spectral properties of the corresponding magnetic
(sub-elliptic) Laplacian. We show that uniform magnetic fields uplift the
bottom of the spectrum. For magnetic fields vanishing at infinity, including
Aharonov--Bohm potentials, we derive magnetic improvements to a variety of
Hardy-type inequalities for the Heisenberg sub-Laplacian. In particular, we
establish a sub-Riemannian analogue of Laptev and Weidl sub-criticality result
for magnetic Laplacians in the plane. Instrumental for our argument is the
validity of a Hardy-type inequality for the Folland--Stein operator, that we
prove in this paper and has an interest on its own.
