  We investigate how the four-dimensional noncommutative open string/Yang-Mills
theory behaves under a general non-perturbative quantum $SL(2,Z)$ symmetry
transformation. We discuss this by considering D3 branes in a constant
background of axion, dilaton, and electric and magnetic fields (including both
$ {\bf E} \perp {\bf B}$ and {\bf E}$||${\bf B} cases) in the respective
decoupling limit. We find that the value of axion, whether rational or
irrational, determines the nature of the resulting theory under $SL(2,Z)$ as
well as its properties such as the coupling constant and the number of
noncommutative directions. In particular, a strongly coupled theory with an
irrational value of axion can never be physically equivalent to a weakly
coupled theory while this is usually true for a theory with a rational value of
axion. A noncommutative Yang-Mills (NCYM) (resulting from D3 branes with pure
magnetic flux) is physically equivalent to a noncommutative open string (NCOS)
but if the value of axion is irrational, we also have noncommutative
space-space directions in addition to the usual noncommutative space-time
directions for NCOS. We also find in general that a NCOS cannot be physically
equivalent to a NCYM but to another NCOS if the value of axion is irrational.
We find another new decoupling limit for possible light-like NCYM whose
$SL(2,Z)$ duality is a light-like ordinary Yang-Mills if the value of the axion
is rational. Various related questions are also discussed.
