Fix the modular curve $X=X_0(N)$ over $\mathbb Q$ and consider also the two cusps induced by the points $0,\infty\in\mathbb P^1(\mathbb Q)$. By abuse of notation let's denote these points on $X_0(N)$ still as $0$ and $\infty$.
Now for any prime $p$ consider the curve $X_p$ which is the reduction modulo $p$ of $X$. Let $0_p,\infty_p\in X_p$ the points induced by $0$ and $\infty$. Is it possible to have $0_p=\infty_p$ for some prime $p$?