  We investigate the Hadamard function and the vacuum expectation value of the
current density for a charged massive scalar field on a slice of anti-de Sitter
(AdS) space described in Poincar\'{e} coordinates with toroidally compact
dimensions. Along compact dimensions periodicity conditions are imposed on the
field with general phases. Moreover, the presence of a constant gauge field is
assumed. The latter gives rise to Aharonov-Bohm-like effects on the vacuum
currents. The current density along compact dimensions is a periodic function
of the gauge field flux with the period equal to the flux quantum. It vanishes
on the AdS boundary and, near the horizon, to the leading order, it is
conformally related to the corresponding quantity in Minkowski bulk for a
massless field. For large values of the length of the compact dimension
compared with the AdS curvature radius, the vacuum current decays as power-law
for both massless and massive fields. This behavior is essentially different
from from the corresponding one in Minkowski background, where the currents for
a massive field are suppressed exponentially.
