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

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.