Modular and Shimura curves $X_0^D(N)$ embedded in $\mathbb P^1 \times \mathbb P^1$

Abstract

We classify the modular curves $X_0(N)$ that admit a smooth embedding in $\mathbb P^1 \times \mathbb P^1$ and obtain a partial classification for the Shimura curves $X_0^D(N)$. We also prove that, for $D>1$, a Shimura curve $X_0^D(N)$ admits a smooth plane model if and only if its genus is at most one.