  It is shown that a supersymmetric and $\kappa$-symmetric D3-brane action on
$AdS_5 \times S^5$ is mapped into itself by a duality transformaion, thereby
verifying the $SL(2,Z)$ invariance of the D3-brane action in the $AdS_5 \times
S^5$ background as in the flat background. To this end, we fix the
$\kappa$-symmetry in a gauge which simplifies the classical action in order to
perform an SO(2) rotation of the N=2 spinor index in a manifest way, though
this may not be necessary. This situation is the same as the case of a super
D-string on $AdS_5 \times S^5$ where it was shown that the super D-string
action is transformed to a form of the IIB Green-Schwarz superstring action
with the $SL(2,Z)$ covariant tension in the $AdS_5 \times S^5$ background
through a duality transformation. These results strongly suggest that various
duality relations originally found in the flat background may be independent of
background geometry, in other words, the duality transformations in string and
p-brane theories may exist even in general curved space-time.
