论文标题
关于表面和Belyi函数的等边三角剖分的一些评论
Some remarks on equilateral triangulations of surfaces and Belyi functions
论文作者
论文摘要
在本文中，在贝利斯（Blethendieck）{\ it esquisse d'un程序}之后，这是由Belyi的工作激发的，我们研究了表面$ X $的一些属性，这些属性是由（可能是理想的）等距等同的三角形三角形的，这是球形，伊氏，伊替体或超质量差异的三角形。这些表面具有圆锥奇点的天然riemannian度量。在欧几里得的情况下，我们分析了封闭的大地测量学及其长度。可以将这样的表面赋予Riemann表面的结构，该表面被视为代数曲线，被Belyi定理在$ \ bar {\ mathbb {q}} $上定义。当然，许多作者对他们进行了研究。在这里，我们定义了两个Belyi函数的连接总和的概念，并提供了一些具体的例子。在特殊情况下，当$ x $是圆环时，三角剖分会导致椭圆曲线，我们定义了从三角剖分（这是橙皮的隐喻）获得的“ peel”的概念，并将其与椭圆曲线的模量$τ$相关。关于椭圆曲线的模块化和taniyama-shimura-weil理论的几何方面出现了许多有趣的问题。
In this paper, following Grothendieck {\it Esquisse d'un programme}, which was motivated by Belyi's work, we study some properties of surfaces $X$ which are triangulated by (possibly ideal) isometric equilateral triangles of one of the spherical, euclidean or hyperbolic geometries. These surfaces have a natural Riemannian metric with conic singularities. In the euclidean case we analyze the closed geodesics and their lengths. Such surfaces can be given the structure of a Riemann surface which, considered as algebraic curves, are defined over $\bar{\mathbb{Q}}$ by a theorem of Belyi. They have been studied by many authors of course. Here we define the notion of connected sum of two Belyi functions and give some concrete examples. In the particular case when $X$ is a torus, the triangulation leads to an elliptic curve and we define the notion of a "peel" obtained from the triangulation (which is a metaphor of an orange peel) and relate this peel with the modulus $τ$ of the elliptic curve. Many fascinating questions arise regarding the modularity of the elliptic curve and the geometric aspects of the Taniyama-Shimura-Weil theory.
