This article explores the connection between radical isogenies and modular curves. Radical isogenies are formulas designed for the computation of chains of isogenies of fixed small degree $N$, introduced by Castryck, Decru, and Vercauteren at Asiacrypt 2020. One significant advantage of radical isogeny formulas over other formulas with a similar purpose is that they eliminate the need to generate a point of order $N$ that generates the kernel of the isogeny. While radical isogeny formulas were originally developed using elliptic curves in Tate normal form, Onuki and Moriya have proposed radical isogeny formulas of degrees $3$ and $4$ on Montgomery curves and attempted to obtain a simpler form of radical isogenies using enhanced elliptic and modular curves. In this article, we translate the original setup of radical isogenies in Tate normal form into the language of modular curves. Additionally, we solve an open problem introduced by Onuki and Moriya regarding radical isogeny formulas on $X_0(N).$

翻译：本文探讨了激进同源与模曲线之间的联系。激进同源是由Castryck、Decru和Vercauteren在Asiacrypt 2020会议上提出的一组公式，专门用于计算固定小次数$N$的同源链。与其他具有类似目的的公式相比，激进同源公式的一个显著优势在于，它无需生成一个阶为$N$的点来定义同源的核。尽管激进同源公式最初是在Tate标准型椭圆曲线上推导出来的，但Onuki和Moriya已在Montgomery曲线上提出了次数为$3$和$4$的激进同源公式，并试图通过增强椭圆曲线和模曲线来获得更简洁的激进同源形式。在本文中，我们将Tate标准型中激进同源的原始设置转化为模曲线的语言。此外，我们还解决了Onuki和Moriya提出的关于$X_0(N)$上激进同源公式的一个开放性问题。