On primitive elements of algebraic function fields and models of $X_0(N)$

TL;DR

This paper investigates the use of primitive elements in finite separable field extensions to analyze maps from modular curves $X_0(N)$ into the projective plane, establishing conditions for birationality and simplicity of the resulting equations.

Contribution

It introduces a new approach based on primitive elements to determine when maps from $X_0(N)$ to $ ext{P}^2$ are birational and produce simple equations.

Findings

Most constructed maps are birational.

Identifies conditions for simplest equations of the image.

Advances understanding of modular curve embeddings.

Abstract

This paper is a continuation of our previous works where we study maps from $X_0(N)$, $N \ge 1$, into $\mathbb P^2$ constructed via modular forms of the same weight and criteria that such a map is birational (see [12]). In the present paper our approach is based on the theory of primitive elements in finite separable field extensions. We prove that in most of the cases the constructed maps are birational, and we consider those such that the resulting equation of the image in $\mathbb P^2$ is simplest possible.