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 $z⟼(f(z):g(z):h(z))$ 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 $X_0(N)$).

arXiv commentIn this version we correct Lemma 5.2 (in old version, now Lemma 5.3) making the correct formula for the total degree of the classical modular polynomial (see Theorem 1.1 in new version). We reworked the introduction a little bit and Section 5 where we compute the degree of classical modular polynomial