有限单值的广义Lamé方程
Generalized Lamé equation with finite monodromy
摘要：
本文研究了具有有限个单值投射群的对称广义Lamé方程的代数形式。特别是，我们考虑方程的$3$正则奇点上的平坦环面$T$的形式\begin{equation*} \begin{split} \frac{d^2 y}{dz^2}-\left[n_1(n_1+1)(\wp(z+a) +\wp(z-a))\right.\left. +A_1(ζ(z+a) - ζ(z-a))+n_0(n_0+1) \wp(z)+B\right]y=0, \end{split} \end{equation*}其中$n_1, n_0 \in \Bbb R$，$A_1,B \in \Bbb C$，和$\wp$是维尔斯特拉斯椭圆函数。我们给出了代数形式上的有限投射单值群的所有群类型的完整列表，并给出了相应的参数$n_0$和$n_1$。对于只有$1$或$2$正则奇点的方程，我们进一步确定了它们的单值群类型。使用的主要工具是Grothendieck对应，它给出了Belyi对和dessin d'enfants之间的双射。利用Klein定理，我们可以把具有有限单值性的广义Lamé方程看作是超几何方程的拉回.在本文中，我们将限制我们的情况下，假设回调映射是Belyi函数。在这种设置下，我们的主要成果包括一个系统的建设所需的设计。特别是粘合程序的dessin将被开发，使感应结构的dessin。
英文摘要：
In this paper, we study the algebraic form of the symmetric generalized Lamé equations which have finite projective monodromy groups. In particular, we consider equations with $3$ regular singular points on a flat torus $T$ which takes the form \begin{equation*} \begin{split} \frac{d^2 y}{dz^2}-\left[n_1(n_1+1)(\wp(z+a) +\wp(z-a))\right.\left. +A_1(ζ(z+a) - ζ(z-a))+n_0(n_0+1) \wp(z)+B\right]y=0, \end{split} \end{equation*} where $n_1, n_0 \in \Bbb R$, $A_1,B \in \Bbb C$, and $\wp$ is the Weierstrass elliptic function. We give a complete list of all the group types that occur as the finite projective monodromy groups on the algebraic form and give the corresponding parameters $n_0$ and $n_1$. For equations with only $1$ or $2$ regular singular points, we further determine their monodromy group types. The main tool used is the Grothendieck correspondence which gives a bijection between Belyi pairs and dessin d'enfants. By Klein's theorem, we may regard generalized Lamé equations with finite monodromy as pullbacks of the hypergeometric equations. In this paper, we will restrict our cases with the assumption that the pullback maps are Belyi functions. Under this setup, our main results consist of a systematic construction on the required dessin. In particular a gluing procedure of dessin will be developed to enable inductive constructions of the dessin.
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