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In this work, we study the Aharonov-Casher effect in the $(2+1)$-dimensionalDirac oscillator coupled to an external electromagnetic field. We set up oursystem in two different scenarios: in the Minkowski spacetime and the cosmicstring spacetime. In both cases, we solve exactly the Dirac oscillator and wedetermine the energy spectrum and the eigenfunctions for the bound states. Weverify that in the Minkowski spacetime, the Dirac oscillator spectrum dependslinearly on the strength of the magnetic field $B$, and on the Aharonov-Casherphase. In addition, we explicitly obtain the corrections on the Dirac spinorsand the energy levels due to the curvature effects in the cosmic stringbackground. Finally, we investigate the nonrelativistic limit and compare ourresults with those found in the literature.

Exact solutions of the Dirac oscillator under the influence of the Aharonov-Casher effect in the cosmic string background