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

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.

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.

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

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

[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). | Zbl | MR

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

[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

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

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

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

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

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

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

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

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

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