The geometry of generalized Lam\'{e} equation, II: Existence of pre-modular forms and application

In this paper, the second in a series, we continue to study the generalized Lam\'{e} equation with the Treibich-Verdier potential \begin{equation*} y^{\prime \prime }(z)=\bigg[ \sum_{k=0}^{3}n_{k}(n_{k}+1)\wp(z+\tfrac{ \omega_{k}}{2}|\tau)+B\bigg] y(z),\quad n_{k}\in \mathbb{Z}_{\geq0} \end{equation*} from the monodromy aspect. We prove the existence of a pre-modular form $Z_{r,s}^{\mathbf{n}}(\tau)$ of weight $\frac{1}{2}\sum n_k(n_k+1)$ such that the monodromy data $(r,s)$ is characterized by $Z_{r,s}^{\mathbf{n}}(\tau)=0$. This generalizes the result in \cite{LW2}, where the Lam\'{e} case (i.e. $n_1=n_2=n_3=0$) was studied by Wang and the third author. As applications, we prove among other things that the following two mean field equations $$\Delta u+e^u=16\pi\delta_{0}\quad\text{and}\quad \Delta u+e^u=8\pi\sum_{k=1}^3\delta_{\frac{\omega_k}{2}}$$ on a flat torus $E_{\tau}:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau)$ has the same number of even solutions. This result is quite surprising from the PDE point of view.