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?



Regular minimal model of $X_0(p^2)$
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](http://www.numdam.org/item?id=AIF_1990__40_1_31_0).


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.


