Radical isogenies and modular curves

This article explores the connection between radical isogenies and modular curves. Radical isogenies are formulas introduced by Castryck, Decru, and Vercauteren at Asiacrypt 2020, designed for the computation of chains of isogenies of fixed small degree $N.$ An important advantage of radical isogeny formulas over other formulas with a similar purpose, is that there is no need to generate a point of order $N$ that generates the kernel of the isogeny. Radical isogeny formulas were originally developed using elliptic curves in Tate normal form, while Onuki and Moriya have proposed radical isogenies formulas of degrees $3$ and $4$ on Montgomery curves. Furthermore, they 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 (using Tate normal form) to the language of modular curves. In addition, we solve an open problem introduced by Onuki and Moriya regarding radical isogeny formulas on $X_0(N).$