Regular minimal model of $X_0(p^2)$

Question

Consider the compactified modular curve $X_0(p^2)$ and the corresponding algebraic curve over $\mathbb{Q}$. My questions are the following:

1. 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?

2. 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?

3. What is a good reference for this material?

Answer 1

The cusp divisor of $X_0(N)$ is given (at least in principle) by the “torsion” of the degenerate elliptic curves and (in principle) described as such in Katz-Mazur's book, Arithmetic moduli of elliptic curves, Annals of math. studies, vol. 108 (1985), Princeton Univ. Press.

An answer to question 2 (and much more) can be found in the paper “Minimal resolution and stable reduction of modular curves”, by Bas Edixhoven published in Annales de l'Institut Fourier, tome 40, n° 1 (1990), p. 31-67. It can be downloaded from Numdam.

By the way, more sophisticated methods allowed Jared Weinstein to compute a stable reduction of general modular curves. See “Semistable models for modular curves of arbitrary level”, Inventiones Math., to appear (and arXived).

These 3 excellent references (read in that order) probably furnish the best way to get informed on the arithmetic properties of modular curves.