  We derive a closed expression for the vacuum expectation value (VEV) of the
fermionic current density in a (D+1)-dimensional locally AdS spacetime with an
arbitrary number of toroidally compactified Poincare spatial dimensions and in
the presence of a constant gauge field. The latter can be formally interpreted
in terms of a magnetic flux treading the compact dimensions. In the compact
subspace, the field operator obeys quasiperiodicity conditions with arbitrary
phases. The VEV of the charge density is zero and the current density has
nonzero components along the compact dimensions only. They are periodic
functions of the magnetic flux with the period equal to the flux quantum and
tend to zero on the AdS boundary. Near the horizon, the effect of the
background gravitational field is small and the leading term in the
corresponding asymptotic expansion coincides with the VEV for a massless field
in the locally Minkowski bulk. Unlike the Minkowskian case, in the system
consisting an equal number of fermionic and scalar degrees of freedom, with
same masses, charges and phases in the periodicity conditions, the total
current density does not vanish. In these systems, the leading divergences in
the scalar and fermionic contributions on the horizon are canceled and, as a
consequence of that, the charge flux, integrated over the coordinate
perpendicular to the AdS boundary, becomes finite. We show that in odd
spacetime dimensions the fermionic fields realizing two inequivalent
representations of the Clifford algebra and having equal phases in the
periodicity conditions give the same contribution to the VEV of the current
density. Combining the contributions from these fields, the current density in
odd-dimensional C-,P- and T -symmetric models are obtained. As an application,
we consider the ground state current density in curved carbon nanotubes.
