黎曼曲面Riemann Surface
A Riemann surface is a surface-like configuration that covers the complex plane with several, and in general infinitely many, "sheets." These sheets can have very complicated structures and interconnections (Knopp 1996, pp. 98-99). Riemann surfaces are one way of representing multiple-valued functions; another is branch cuts. The above plot shows Riemann surfaces for solutions of the equation
黎曼曲面是一种类似于曲面的结构，它覆盖了多个，通常是无限多个的“片”。这些片可以有非常复杂的结构和相互连接（Knopp 1996，pp.98-99）。Riemann曲面是表示多值函数（功能）的一种方法；另一种是分支切割。上图显示了方程解的黎曼曲面。
其中d=2, 3, 4, and 5, where w(z) is the Lambert W-function (M. Trott).
The Riemann surface S of the function field K is the set of nontrivial
discrete valuations on K. Here, the set S corresponds to the ideals of the ring A of K integers of K over C(z) . ( A consists of the elements
of K that are roots of monic polynomials over C(z) .) Riemann surfaces provide a geometric visualization of functions elements and their analytic
continuations.
函数（功能）域K的Riemann曲面S是K上的一组非平凡离散赋值集，这里的S对应于C（z）上K的整数环A的理想。（A由K的元素组成，这些元素是C[z]上的一元多项式的根）。Riemann曲面提供了函数（功能）元素及其解析连续性的几何可视化。
Schwarz proved at the end of nineteenth century that the automorphism
group of a compact Riemann surface of genus g>=2 is finite, and Hurwitz (1893) subsequently showed that its order is at most 84(g-1) (Arbarello et
al. 1985, pp. 45-47; Karcher and Weber 1999, p. 9). This bound is attained for infinitely many g, with the smallest of g such an extremal surface being 3 (corresponding to the Klein quartic). However, it is also known that there are infinitely many genera for which the bound 84（g-1） is not attained (Belolipetsky 1997, Belolipetsky and Jones).
Schwarz在十九世纪末证明了亏格g>=2的紧致黎曼曲面的自同构群是有限的，Hurwitz（1893）随后证明了它的阶至多为84（g-1）（Arbarello等人。1985年，第45-47页；卡彻和韦伯1999年，第9页）。对于无穷多的g，这个界是得到的，并且这样一个极值曲面的最小g是3（对应于Klein四次曲线）。然而，我们也知道，有无限多的属没有达到84（g-1）的界限（belloipetsky 1997，belloipetsky和Jones）。