On a Theorem of Hurwitz
On a Theorem of Hurwitz
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关于赫尔维茨定理
DOI：
10.1017/s2040618500034365
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发表时间：
1961
期刊：
影响因子：
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通讯作者：
A. Macbeath
中科院分区：
文献类型：
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作者：
A. Macbeath
根据 Hurwitz 定理 [3]，属 g ≥ 2 的代数曲线不能有超过 84(g − l) 个双有理自变换，或者我们称之为自同构。属 3 的克莱因四次方程已达到界限 [4]。在研究是否存在任何其他曲线可以达到界限的问题时，我被引导考虑黎曼曲面的通用覆盖空间，正如西格尔所观察到的，该空间将赫尔维茨定理与西格尔自己关于 Fuchsian 群的基本区域的度量的结果 [7] 联系起来。任何具有 84(g − 1) 个自同构的曲线必须由三角形群 (2, 3, 7) 的正规子群统一化，并且通过对 (2, 3, 7) 可能的有限因子群进行更仔细的分析，纯代数方法会产生具有最大自同构数的无限曲线族。这将在稍后的论文中展示。
By a theorem of Hurwitz [3], an algebraic curve of genus g ≧ 2 cannot have more than 84(g − l) birational self-transformations, or, as we shall call them, automorphisms. The bound is attained for Klein's quartic of genus 3 [4]. In studying the problem whether there are any other curves for which the bound is attained, I was led to consider the universal covering space of the Riemann surface, which, as Siegel observed, relates Hurwitz's theorem to Siegel's own result [7] on the measure of the fundamental region of Fuchsian groups. Any curve with 84(g − 1) automorphisms must be uniformized by a normal subgroup of the triangle group (2, 3, 7), and, by a closer analysis of possible finite factor groups of (2, 3, 7), purely algebraic methods yield an infinite family of curves with the maximum number of automorphisms. This will be shown in a later paper.