# 从模函数到单值化定理 Ⅳ

Prologue:

…At the moment when I put my foot on the step the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuchsian functions were identical with those of non-Euclidean geometry. I did not verify the idea; I should not have had time, as, upon taking my seat in the omnibus, I went on with a conversation already commenced, but I felt a perfect certainty…

——H. Poincaré

H. Poincaré (1854-1912)

$\mathbb{\bar{C}}$的解析自同构群为Möbius变换群，等价的说法是$\mathbb{C}P^{1}$的解析自同构群为$PSL(2,\mathbb{C})$$PSL(2,\mathbb{C})$中所有保持无穷远点不动的元素有形式$\left({\begin{array}{cc} a&b\\ 0&d\\ \end{array}}\right)$，此即$\mathbb{C}$上的线性函数，在迭代下构成$\mathbb{C}$的解析自同构群。最后，Schwarz引理的一个经典应用定出$\triangle$的解析自同构有形式$e^{i\theta}(z-z_{0})/(1-\bar{z_{0}}z)$$\theta \in \mathbb{R}$$z_{0} \in \triangle$

a) 以$\mathbb{\bar{C}}$为万有覆叠的Riemann面。由于$PSL(2,\mathbb{C})$中的元素作用在$\mathbb{C}P^{1}$上恒有不动点，推出$G$是平凡的。故唯一以$\mathbb{\bar{C}}$为万有覆叠的Riemann面是$\mathbb{\bar{C}}$本身。

b) 以$\mathbb{C}$为万有覆叠的Riemann面。忠实作用在$\mathbb{C}$上的线性变换有形式$z \mapsto z+b$。取平移函数作为$G$的生成元，并要求其在$\mathbb{Q}$上线性无关。离散群$G$至多有2个这样的生成元，分类讨论得到$S$解析同构于$\mathbb{C}$$\mathbb{C}\backslash\{0\}$（借助一个指数变换）或$\mathbb{C}/\Lambda$$\Lambda$是某个格。

c)除了以上讨论过的4种例外，其余 Riemann面均以$\triangle$$\mathbb{H}$）为万有覆叠。这是最为有趣的情形，因为我们缺少$PSL(2,\mathbb{R})$$\mathbb{H}$上忠实作用的离散子群的完备描述。Poincaré第一个研究了$PSL(2,\mathbb{R})$的离散子群，并将它们命名为Fuchs群。如果Fuchs群的所有元素都忠实地作用在$\mathbb{H}$上，则称其为无挠的。显然，以$\mathbb{H}$为万有覆叠的Riemann面同构于$\mathbb{H} /G$$G$是某个无挠的Fuchs群。

Many other proofs have been given which are more elementary in that they need less preparation, but none is as penetrating as the original proof.