Structure of critical points of multiple Green functions on flat torus and applications

Let $E_τ:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}τ)$ be a flat torus and $G(z)=G(z;τ)$ be the Green function on $E_τ$ with the singularity at $0$. Consider the multiple Green function $G_{n}$ on $(E_τ)^{n}$: \[ G_{n}(z_{1},\cdots,z_{n}):=\sum_{i<j}G(z_{i}-z_{j})-n\sum_{i=1}^{n}G(z_{i}). \] A critical poin