Minimal resolution and stable reduction of X 0 (N)
Annales de l'Institut Fourier, Tome 40 (1990) no. 1, pp. 31-67.

Soit N1 un nombre entier. Soit X 0 (N) la courbe modulaire sur Z, construite par Katz et Mazur. On calcule la résolution minimale de X 0 (N) sur Z[1/6]. Soit p5 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 (N) a réduction stable en p sur Q[p n], et on calcule la fibre au-dessus de p du modèle stable.

Let N1 be an integer. Let X 0 (N) be the modular curve over Z, as constructed by Katz and Mazur. The minimal resolution of X 0 (N) over Z[1/6] is computed. Let p5 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 (N) has stable reduction at p over Q[p n], and the fibre at p of the stable model is computed.

@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},
     pages = {31--67},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {40},
     number = {1},
     year = {1990},
     doi = {10.5802/aif.1202},
     mrnumber = {92f:11080},
     zbl = {0679.14009},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1202/}
}
TY  - JOUR
AU  - Edixhoven, Bas
TI  - Minimal resolution and stable reduction of $X_0(N)$
JO  - Annales de l'Institut Fourier
PY  - 1990
SP  - 31
EP  - 67
VL  - 40
IS  - 1
PB  - Institut Fourier
PP  - Grenoble
UR  - http://www.numdam.org/articles/10.5802/aif.1202/
DO  - 10.5802/aif.1202
LA  - en
ID  - AIF_1990__40_1_31_0
ER  - 
%0 Journal Article
%A Edixhoven, Bas
%T Minimal resolution and stable reduction of $X_0(N)$
%J Annales de l'Institut Fourier
%D 1990
%P 31-67
%V 40
%N 1
%I Institut Fourier
%C Grenoble
%U http://www.numdam.org/articles/10.5802/aif.1202/
%R 10.5802/aif.1202
%G en
%F AIF_1990__40_1_31_0
Edixhoven, Bas. Minimal resolution and stable reduction of $X_0(N)$. Annales de l'Institut Fourier, Tome 40 (1990) no. 1, pp. 31-67. doi : 10.5802/aif.1202. http://www.numdam.org/articles/10.5802/aif.1202/

[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). | MR | Zbl

[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 | Zbl

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

[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). | Numdam | MR | Zbl

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

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

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

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

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

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

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

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

Cité par Sources :