The geometry of generalized Lame equation, III: One-to-one of the Riemann-Hilbert correspondence

Abstract

In this paper, the third in a series, we continue to study the generalized Lam\'{e} equation H$(n_0,n_1,n_2,n_3;B)$ with the Darboux-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*} and a related linear ODE with additional singularities $\pm p$ from the monodromy aspect.We establish the uniqueness of these ODEs with respect to the global monodromy data. Surprisingly, our result shows that the Riemann-Hilbert correspondence from the set \[\{\text{H}(n_0,n_1,n_2,n_3;B)|B\in\mathbb{C}\}\cup \{\text{H}(n_0+2,n_1,n_2,n_3;B) | B\in\mathbb{C}\}\] to the set of group representations $\rho:\pi_1(E_{\tau})\to SL(2,\mathbb{C})$ is one-to-one. We emphasize that this result is not trivial at all. There is an example that for $\tau=\frac12+i\frac{\sqrt{3}}{2}$, there are $B_1,B_2$ such that the monodromy representations of H$(1,0,0,0;B_1)$ and H$(4,0,0,0;B_2)$ are {\bf the same}, namely the Riemann-Hilbert correspondence from the set \[\{\text{H}(n_0,n_1,n_2,n_3;B)|B\in\mathbb{C}\}\cup \{\text{H}(n_0+3,n_1,n_2,n_3;B) | B\in\mathbb{C}\}\] to the set of group representations is {\bf not} necessarily one-to-one. This example shows that our result is completely different from the classical one concerning linear ODEs defined on $\mathbb{CP}^1$ with finite singularities.