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
黎曼曲面是一種類似於曲面的結構，它覆蓋了多個，通常是無限多個的“片”。這些片可以有非常復雜的結構和相互連接（Knopp 1996，pp.98-99）。Riemann曲面是表示多值函數（功能）的一種方法；另一種是分支切割。上圖顯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.
函數（功能）域K的Riemann曲面S是K上的一組非平凡離散賦值集，這里的S對應於C（z）上K的整數環A的理想。（A由K的元素組成，這些元素是C[z]上的一元多項式的根）。Riemann曲面提供了函數（功能）元素及其解析連續性的幾何可視化。
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在十九世紀末證明了虧格g>=2的緊致黎曼曲面的自同構群是有限的，Hurwitz（1893）隨后證明了它的階至多為84（g-1）（Arbarello等人。1985年，第45-47頁；卡徹和韋伯1999年，第9頁）。對於無窮多的g，這個界是得到的，並且這樣一個極值曲面的最小g是3（對應於Klein四次曲線）。然而，我們也知道，有無限多的屬沒有達到84（g-1）的界限（belloipetsky 1997，belloipetsky和Jones）。