Integral Models of $X_0(N)$ and Their Degrees

TL;DR

This paper computes degrees of models of modular curves using modular forms, establishing birational maps, and provides explicit models with degrees related to classical functions, enhancing understanding of modular curve embeddings.

Contribution

It introduces explicit integral models of $X_0(N)$ with degrees tied to the Dedekind psi function, and analyzes their properties using modular forms of specific weights.

Findings

Models of $X_0(N)$ have degree $ o \psi(N)$ with integral $q$-expansions.

Existence of models with degree $ o rac{ ext{psi}(N)}{3}$ using cuspidal forms.

Degree formulas for classical modular polynomials and canonical models.

Abstract

In this paper we compute the degree of a curve which is the image of a mapping $z\longmapsto (f(z): g(z): h(z))$ constructed out of three linearly independent modular forms of the same even weight $\ge 4$ into $\mathbb P^2$. We prove that in most cases this map is a birational equivalence and defined over $\mathbb Z$. We use this to construct models of $X_0(N)$, $N\ge 2$, using modular forms in $M_{12}(\Gamma_0(N))$ with integral $q$--expansion. The models have degree equal to $\psi(N)$ (a classical Dedekind psi function). When genus is at least one, we show the existence of models constructed using cuspidal forms in $S_4(\Gamma_0(N))$ of degree $\le \psi(N)/3$ and in $S_6(\Gamma_0(11))$ of degree 4. As an example of a different kind, we compute the formula for the total degree i.e., the degree considered as a polynomial of two (independent) variables of the classical modular polynomial (or the degree of the canonical model of $X_0(N)$).