The geometry of generalized Lamé equation, II: Existence of pre-modular forms and application

Abstract

In this paper, the second in a series, we continue to study the generalized Lamé 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{ ω_{k}}{2}|τ)+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 Lamé 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 \[Δu+e^u=16πδ_{0}\quad\text{and}\quad Δu+e^u=8π\sum_{k=1}^3δ_{\frac{ω_k}{2}}\] 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.

23pages