in this talk i would like to discuss the global rigidity property for local holomorphic maps from an open piece of a hermitian symmetric space $m$ into a cartesian product of $m$. this study has been related to problems in number theory in classifying the modular correspondences, as initated by the work of clozel-ullmo. we will discuss the work of mok-ng on the rigidity pehomenon when the map is local area preserving and $m$ is of non-compact type. we then focus on our recent joint work with h. fang and m. xiao when $m$ is of compact type.