论文标题
圆环bely \uı地图的关键点
Critical Points of Toroidal Bely\uı Maps
论文作者
论文摘要
belyi map $β：\ mathbb {p}^1（\ mathbb {c}）\ to \ m athbb {p}^1（\ mathbb {c}）$是一个合理函数，最多为三个关键值；我们可以假设这些值为$ \ {0，\，1，\，\ infty \} $。用椭圆曲线$ e替换$ \ mathbb {p}^1 $：\ y^2 = x^3 + a \，x + b $，也有类似的定义Belyi Map $β：e（\ Mathbb {C}）由于$ e（\ mathbb {c}）\ simeq \ mathbb t^2（\ mathbb {r}）$是圆环，我们称$（e，β）$为toroidal belyi对。 Belyi Maps $β：E（\ Mathbb {C}）\ to \ Mathbb p^1（\ Mathbb {C}）$有很多示例；可以在LMFDB在线找到一些。给定这样的圆环belyi度$ n $的图像，逆图$ g =β^{ - 1} \ bigl（\ {0，\，1，\，\，\ infty \} \ bigr）$是$ n $元素的集合，其中包含Belyi Map的关键点。在此项目中，我们调查$ e（\ Mathbb {c}）_ {\ text {tors}} $中包含$ g $。这是作为Pomona数学经验研究（NSA H98230-21-1-0015）的一部分完成的工作。
A Belyi map $β: \mathbb{P}^1(\mathbb{C}) \to \mathbb{P}^1(\mathbb{C})$ is a rational function with at most three critical values; we may assume these values are $\{ 0, \, 1, \, \infty \}$. Replacing $\mathbb{P}^1$ with an elliptic curve $E: \ y^2 = x^3 + A \, x + B$, there is a similar definition of a Belyi map $β: E(\mathbb{C}) \to \mathbb{P}^1(\mathbb{C})$. Since $E(\mathbb{C}) \simeq \mathbb T^2(\mathbb {R})$ is a torus, we call $(E, β)$ a Toroidal Belyi pair. There are many examples of Belyi maps $β: E(\mathbb{C}) \to \mathbb P^1(\mathbb{C})$ associated to elliptic curves; several can be found online at LMFDB. Given such a Toroidal Belyi map of degree $N$, the inverse image $G = β^{-1} \bigl( \{ 0, \, 1, \, \infty \} \bigr)$ is a set of $N$ elements which contains the critical points of the Belyi map. In this project, we investigate when $G$ is contained in $E(\mathbb{C})_{\text{tors}}$. This is work done as part of the Pomona Research in Mathematics Experience (NSA H98230-21-1-0015).
