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.