  We investigate probe limit vortex solutions of a charged scalar field in
Einstein-Maxwell theory in 3+1 dimensions, for an asymptotically AdS
Schwarzschild black hole metric with the addition of an axionic coupling to the
Maxwell field. We show that the inclusion of such a term, together with a
suitable potential for the axion field, can induce an effective Chern-Simons
term on the 2+1 dimensional boundary. We obtain numerical solutions of the
equations of motion and find Maxwell-Chern-Simons like magnetic vortex
configurations, where the magnetic field profile varies with the size of the
effective Chern-Simons coupling. The axion field has a non-trivial profile
inside the AdS bulk but does not condense at spatial infinity.
