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

@article{AIF_1990__40_1_31_0,
author = {Edixhoven, Bas},
title = {Minimal resolution and stable reduction of $X\_0(N)$},
journal = {Annales de l'Institut Fourier},
publisher = {Imprimerie Louis-Jean},
volume = {40},
number = {1},
year = {1990},
pages = {31-67},
doi = {10.5802/aif.1202},
zbl = {0679.14009},
mrnumber = {92f:11080},
language = {en},
url = {http://www.numdam.org/item/AIF_1990__40_1_31_0}
}

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/item/AIF_1990__40_1_31_0/

[1] B.J. Birch and W. Kuyk, Modular functions of one variable IV, Springer Lecture Notes in Mathematics, 476 (1975). | Zbl 0315.14014

[2] P. Deligne and N. Katz, Séminaire de géométrie algébrique 7 II, Springer Lecture Notes in Mathematics, 340 (1973). | MR 50 #7135 | Zbl 0258.00005

[3] P. Deligne and M. Rapoport, Les schémas de modules des courbes elliptiques. In Modular Functions of One Variable II, Springer Lecture Notes in Mathematics, 349 (1973). | MR 49 #2762 | Zbl 0281.14010

[4] B.H. Gross and D.B. Zagier, Heegner points and derivatives of L-series, Invent. Math., 84 (1986), 225-320 | MR 87j:11057 | Zbl 0608.14019

[5] A. Grothendieck, Eléments de géométrie algébrique, Ch. I, II, III, IV, Publications Mathématiques de l'I.H.E.S, 4, 8, 11, 17, 20, 24, 28, 32 (1960-1967). | Numdam

[6] A. Grothendieck, Séminaire de géométrie algébrique I : Revêtements étales et groupe fondamental, Springer Lecture Notes in Mathematics, 224 (1971). | Zbl 0234.14002

[7] R. Hartshorne, Curves with high selfintersection on algebraic surfaces, Publications Mathématiques de l'I.H.E.S, 36 (1969). | Numdam | MR 42 #1826 | Zbl 0197.17505

[8] R. Hartshorne, Algebraic geometry, Springer Graduate Texts in Mathematics, 52 (1977). | MR 57 #3116 | Zbl 0367.14001

[9] N.M. Katz and B. Mazur, Arithmetic moduli of elliptic curves, Annals of Mathematics Studies, Princeton University Press, 108 (1985). | MR 86i:11024 | Zbl 0576.14026

[10] J. Lipman, Rational singularities, with applications to algebraic surfaces and unique factorization, Publications Mathématiques de l'I.H.E.S, 36 (1969). | Numdam | MR 43 #1986 | Zbl 0181.48903

[11] B. Mazur, Modular curves and the Eisenstein ideal, Publications Mathématiques de l'I.H.E.S, 47 (1977), 33-186. | Numdam | MR 80c:14015 | Zbl 0394.14008

[12] J.F. Mestre, Courbes de Weil et courbes supersingulières, Séminaire de théorie des nombres 1984-1985, Université de Bordeaux 1. | Zbl 0599.14031

[13] A. Pizer, An algorithm for computing modular forms on Γ0(N), Journal of Algebra, 64 (1980), 340-390. | MR 83g:10020 | Zbl 0433.10012

[14] A. Pizer, Theta series and modular forms of level p2M, Compositio Math., 40 (1980), 177-241. | Numdam | MR 81k:10040 | Zbl 0411.10007 | Zbl 0416.10021

[15] J.-P. Serre, Colloque d'algèbre, 6-7 mai 1967, ENSJF.