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

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 $X_0(N)$).