Minimal resolution and stable reduction of $X_0\left(N\right)$

Edixhoven, Bas

Annales de l'Institut Fourier, Tome 40 (1990) no. 1, pp. 31-67.

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 $\mathbf{Z}[1/6]$. Soit $p\ge 5$ un nombre premier, tel que $N=p^{2}M$, avec $M$ premier à $p$. Soit $n=(p^2-1)/2$. 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.

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 $\mathbf{Z}[1/6]$ is computed. Let $p\ge 5$ be a prime, such that $N=p^{2}M$, with $M$ prime to $p$. Let $n=(p^2-1)/2$. 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.

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