  We discuss the action of SL(2,Z) on local operators in D=4, N=4 SYM theory in
the superconformal phase. The modular property of the operator's scaling
dimension determines whether the operator transforms as a singlet, or
covariantly, as part of a finite or infinite dimensional multiplet under the
SL(2,Z) action. As an example, we argue that operators in the Konishi multiplet
transform as part of a (p,q) PSL(2,Z) multiplet. We also comment on the
non-perturbative local operators dual to the Konishi multiplet.
