Stable reduction of three point covers

Wewers, Stefan

Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 1, pp. 405-421.

Cette note est un survol des résultats récents sur la réduction semi-stable des revêtements de la droite projective ramifiés en trois points.

This note gives a survey of some recent results on the stable reduction of covers of the projective line branched at three points.

[1] A. Abbes, Réduction semi–stable des courbes d’après Artin, Deligne, Grothendieck, Mumford, Saito, Winters,.... Courbes semi-stables et groupe fondamental en géometrie algébrique (J.-B. Bost, F. Loeser, and M. Raynaud, eds.), Prog. in Math., vol. 187, Birkhäuser, 2000, 59–108. | MR | Zbl

[2] S. Beckmann, Ramified primes in the field of moduli of branched coverings of curves. J. of Algebra 125 (1989), 236–255. | MR | Zbl

[3] G.V. Belyi, On Galois extensions of a maximal cyclotomic field. Math. USSR Izw. 14 (1980), no. 2, 247–256. | MR | Zbl

[4] I.I. Bouw, S. Wewers, Stable reduction of modular curves. Modular curves and abelian varieties (J. Cremona et.al., ed.), Progress in Math., no. 224, Birkhäuser, 1–22. | MR | Zbl

[5] R.F. Coleman, On the components of $X_0\left(p^n\right)$. J. Number Theory 110 (2005), no. 1, 3–21. | MR | Zbl

[6] P. Dèbes, J.-C. Douai, Algebraic covers: Field of moduli versus field of definition. Ann. Sci. École Norm. Sup. 30 (1997), 303–338. | Numdam | MR | Zbl

[7] P. Deligne, D. Mumford, The irreducibility of the space of curves of given genus. Publ. Math. IHES 36 (1969), 75–109. | Numdam | MR | Zbl

[8] P. Deligne, M. Rapoport, Les schémas de modules de courbes elliptiques. Modular functions of one variable II, LNM, no. 349, Springer-Verlag, 1972, 143–316. | MR | Zbl

[9] B. Edixhoven, Minimal resolution and stable reduction of $X_0\left(N\right)$. Ann. Inst. Fourier 40 (1990), no. 1, 31–67. | Numdam | MR | Zbl

[10] A. Grothendieck, Revêtement étales et groupe fondamental (SGA I). LNM, no. 224, Springer–Verlag, 1971. | MR

[11] Y. Henrio, Arbres de Hurwitz et automorphismes d’ordre $p$ des disques et des couronnes $p$-adic formels. To appear in: Compositio Math., available at arXiv:math.AG/0011098, 2000.

[12] —, Disque et couronnes ultramétriques. Courbes semi-stables et groupe fondamental en géometrie algébrique (J.-B. Bost, F. Loeser, and M. Raynaud, eds.), Prog. in Math., vol. 187, Birkhäuser, 2000, 7–18. | MR

[13] B. Köck, Belyi’s theorem revisited. arXiv:math.AG/0108222.

[14] C. Lehr, M. Matignon, Wild monodromy and automorphisms of curves. Proceedings of the 2003 Workshop on Cryptography and Related Mathematics, Chuo University, 2003, available at http://www.math.u-bordeaux.fr/~matignon.

[15] Q. Liu, Reduction and lifting of finite covers of curves. Proceedings of the 2003 Workshop on Cryptography and Related Mathematics, Chuo University, 2003, available at http://www.math.u-bordeaux.fr/liu, 161–180.

[16] G. Malle, B. H. Matzat, Inverse Galois theory. Monographs in Mathematics, Springer, 1999. | MR | Zbl

[17] F. Orgogozo, I. Vidal, Le théorèm de spécialisation du groupe fondamental. Courbes semi-stables et groupe fondamental en géometrie algébrique (J.-B. Bost, F. Loeser, and M. Raynaud, eds.), Prog. in Math., vol. 187, Birkhäuser, 2000, 169–184. | MR | Zbl

[18] M. Raynaud, $p$-groupes et réduction semi-stable des courbes. Grothendieck Festschrift III (P. Cartier, ed.), Progress in Math., no. 88, Birkhäuser, 1990, 179–197. | MR | Zbl

[19] —, Revêtement de la droite affine en caractéristique $p>0$ et conjecture d’Abhyankar. Invent. Math. 116 (1994), 425–462. | MR | Zbl

[20] —, Spécialisation des revêtements en caractéristique $p>0$. Ann. Sci. École Norm. Sup. 32 (1999), no. 1, 87–126. | Numdam | MR

[21] L. Schneps (ed.), The grothendieck theory of dessins d’enfants. London Math. Soc. Lecture Note Series, no. 200, Cambridge Univ. Press, 1994. | MR | Zbl

[22] H. Völklein, Groups as Galois groups. Cambridge Studies in Adv. Math., no. 53, Cambridge Univ. Press, 1996. | MR | Zbl

[23] S. Wewers, Formal deformations of curves with group scheme action. arXiv:math.AG/0212145, to appear in: Ann. Inst. Fourier. | Numdam | MR | Zbl

[24] —, Deformation of tame admissible covers of curves. Aspects of Galois theory (H. Völklein, ed.), London Math. Society Lecture Notes Series, no. 256, Cambridge Univ. Press, 1999, 239–282. | MR | Zbl

[25] —, Reduction and lifting of special metacyclic covers. Ann. Sci. École Norm. Sup. 36 (2003), 113–138. | Numdam | MR | Zbl

[26] —, Three point covers with bad reduction. J. Amer. Math. Soc. 16 (2003), no. 4, 991–1032. | MR | Zbl