论文标题
普遍la脚方程的几何形状，iii：riemann-hilbert对应的一对一
The geometry of generalized Lame equation, III: One-to-one of the Riemann-Hilbert correspondence
论文作者
论文摘要
在本文中，我们将继续研究广义的Lamé方程h $（n_0，n_1，n_2，n_3; b）$，以及darboux-treibich-verdier电位\ begin {equation*} y^} y^} \ sum_ {k = 0}^{3} n_ {k}（n_ {k} +1）\ wp（z+\ tfrac {ω__{ω_{k}}} {2} {2} |τ）+b \ bigG] y（z） \ end {equation*}和一个相关的线性极，带有单个奇异性$ \ pm p $从单片方面进行。我们在全球单莫罗莫（Global Monodromy Data）方面确定了这些ODES的独特性。令人惊讶的是，我们的结果表明，riemann-hilbert来自集合\ [\ {\ text {h}（n_0，n_1，n_1，n_2，n_3; b）| b \ in \ in \ mathbb {c} \} \ cup cup \ cup \ cup \ cup { b \ in \ mathbb {c} \} \] to组表示$ρ：π_1（e_τ）\ to sl（2，\ mathbb {c}）$是一对一。我们强调，这个结果根本不是微不足道的。有一个例子是，对于$τ= \ frac12+i \ frac {\ sqrt {3}}} {2} $，有$ b_1，b_2 $，使得h $（1,0,0,0,0,0; b_1; b_1）$和h $（h $ h $（4,0,0,0,0,0,0,0,0; b_ s s s s nemodromy表示） riemann-hilbert来自该集合的通讯\ [\ {\ text {h}（n_0，n_1，n_2，n_2，n_3; b）| b \ in \ mathbb {c} \} \} \ cup \ {\ cup \ {\ text {\ text {h} B \ in \ Mathbb {C} \} \]组为组表示的集合{\ bf not}必然是一对一。该示例表明，我们的结果与经典的结果完全不同，涉及在$ \ mathbb {cp}^1 $上定义的线性ODE，并具有有限的奇异性。
In this paper, the third in a series, we continue to study the generalized Lamé equation H$(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{ ω_{k}}{2}|τ)+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 $ρ:π_1(E_τ)\to SL(2,\mathbb{C})$ is one-to-one. We emphasize that this result is not trivial at all. There is an example that for $τ=\frac12+i\frac{\sqrt{3}}{2}$, 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.
