On a simple model of X_0(N)

Abstract

We find plane models for all $X_0(N)$, $N\ge 2$. We observe a map from the modular curve $X_0(N)$ to the projective plane constructed using modular forms of weight $12$ for the group ${\Gamma }_0(N)$; the Ramanujan function $\Delta$, $\Delta (N\cdot )$ and the third power of Eisestein series of weight $4$, $E_4^3$, and prove that this map is birational equivalence for every $N\ge 2$. The equation of the model is the minimal polynomial of $\Delta (N\cdot )/\Delta$ over $\mathbb{C}(j)$.