  We show that the Green functions on flat tori can have either 3 or 5 critical
points only. There does not seemto be any directmethod to attack this problem.
Instead, we have to employ sophisticated non-linear partial differential
equations to study it.
  We also study the distribution of number of critical points over the moduli
space of flat tori through deformations. The functional equations of special
theta values provide important inequalities which lead to a solution for all
rhombus tori.
