Minimal resolution and stable reduction of ${X}_{0}\left(N\right)$
Annales de l'Institut Fourier, Volume 40 (1990) no. 1, pp. 31-67.

Let $N\ge 1$ be an integer. Let ${X}_{0}\left(N\right)$ be the modular curve over $\mathbf{Z}$, as constructed by Katz and Mazur. The minimal resolution of ${X}_{0}\left(N\right)$ over $\mathbf{Z}\left[1/6\right]$ is computed. Let $p\ge 5$ be a prime, such that $N={p}^{2}M$, with $M$ prime to $p$. Let $n=\left({p}^{2}-1\right)/2$. It is shown that ${X}_{0}\left(N\right)$ has stable reduction at $p$ over $\mathbf{Q}\left[\sqrt[n]{p}\right]$, and the fibre at $p$ of the stable model is computed.

Soit $N\ge 1$ un nombre entier. Soit ${X}_{0}\left(N\right)$ la courbe modulaire sur $\mathbf{Z}$, construite par Katz et Mazur. On calcule la résolution minimale de ${X}_{0}\left(N\right)$ sur $\mathbf{Z}\left[1/6\right]$. Soit $p\ge 5$ un nombre premier, tel que $N={p}^{2}M$, avec $M$ premier à $p$. Soit $n=\left({p}^{2}-1\right)/2$. On montre que ${X}_{0}\left(N\right)$ a réduction stable en $p$ sur $\mathbf{Q}\left[\sqrt[n]{p}\right]$, et on calcule la fibre au-dessus de $p$ du modèle stable.

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title = {Minimal resolution and stable reduction of $X_0(N)$},
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Edixhoven, Bas. Minimal resolution and stable reduction of $X_0(N)$. Annales de l'Institut Fourier, Volume 40 (1990) no. 1, pp. 31-67. doi : 10.5802/aif.1202. http://www.numdam.org/articles/10.5802/aif.1202/

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