Consider the compactified modular curve $X_0(p^2)$ and the corresponding algebraic curve over $\mathbb{Q}$. My questions are the following:
Where do the cusps of $X_0(p^2)_{\mathbb{Q}}$ live? That is for which extension $K$ of $\mathbb{Q}$, $X_0(p^2)(K)$ contains all the cusps?
Let $K$ be as above and $\mathcal{O}_K$ the ring of integers of $K$. Consider the regular minimal model of $X_0(p^2)$ over $\mathcal{O}_K$. What are the singular fibers and what do they look like?
What is a good reference for this material?