In my talk, I will talk about , on one hand,how to use elliptic function theory to construct solutions of a specific mean field equation on torus, when the parameters are integer multiples of 4 pi. On the other hand, the PDE deep theory of bubbling analysis can be applied to obtain results related to the function theory on torus, for example, we can prove the Green function of torus has at most five critical points. Open problems of this aspect is also discussed.