This talk will develope a theory to connect the mean field equation, Green function and the classic Lame equation. In this theory we have constructed a family of hyper-elliptic curves(depending on the genus n and moduli parameter $/taut /in H $: the upper half plane) and a premodular form of degree $/frac 12 n (n + 1)$. We prove that the nonlinear PDE on $E_/tau$ has a solution iff $/tau $ is a zero of this premodular form.