Stable reduction of three point covers

Stefan Wewers[1]

  • [1] Mathematisches Institut Universität Bonn Beringstr. 1 53115 Bonn, BRD

Journal de Théorie des Nombres de Bordeaux (2005)

  • Volume: 17, Issue: 1, page 405-421
  • ISSN: 1246-7405

Abstract

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

How to cite

top

Wewers, Stefan. "Stable reduction of three point covers." Journal de Théorie des Nombres de Bordeaux 17.1 (2005): 405-421. <http://eudml.org/doc/249470>.

@article{Wewers2005,
abstract = {This note gives a survey of some recent results on the stable reduction of covers of the projective line branched at three points.},
affiliation = {Mathematisches Institut Universität Bonn Beringstr. 1 53115 Bonn, BRD},
author = {Wewers, Stefan},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Belyi map; semistable reduction; modular curves},
language = {eng},
number = {1},
pages = {405-421},
publisher = {Université Bordeaux 1},
title = {Stable reduction of three point covers},
url = {http://eudml.org/doc/249470},
volume = {17},
year = {2005},
}

TY - JOUR
AU - Wewers, Stefan
TI - Stable reduction of three point covers
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2005
PB - Université Bordeaux 1
VL - 17
IS - 1
SP - 405
EP - 421
AB - This note gives a survey of some recent results on the stable reduction of covers of the projective line branched at three points.
LA - eng
KW - Belyi map; semistable reduction; modular curves
UR - http://eudml.org/doc/249470
ER -

References

top
  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. Zbl0978.14031MR1768094
  2. S. Beckmann, Ramified primes in the field of moduli of branched coverings of curves. J. of Algebra 125 (1989), 236–255. Zbl0698.14024MR1012673
  3. G.V. Belyi, On Galois extensions of a maximal cyclotomic field. Math. USSR Izw. 14 (1980), no. 2, 247–256. Zbl0429.12004MR534593
  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. Zbl1147.11316MR2058639
  5. R.F. Coleman, On the components of X 0 ( p n ) . J. Number Theory 110 (2005), no. 1, 3–21. Zbl1108.14022MR2114670
  6. P. Dèbes, J.-C. Douai, Algebraic covers: Field of moduli versus field of definition. Ann. Sci. École Norm. Sup. 30 (1997), 303–338. Zbl0906.12001MR1443489
  7. P. Deligne, D. Mumford, The irreducibility of the space of curves of given genus. Publ. Math. IHES 36 (1969), 75–109. Zbl0181.48803MR262240
  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. Zbl0281.14010MR337993
  9. B. Edixhoven, Minimal resolution and stable reduction of X 0 ( N ) . Ann. Inst. Fourier 40 (1990), no. 1, 31–67. Zbl0679.14009MR1056773
  10. A. Grothendieck, Revêtement étales et groupe fondamental (SGA I). LNM, no. 224, Springer–Verlag, 1971. MR354651
  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. MR1768107
  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. Zbl1116.14020
  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. Zbl0940.12001MR1711577
  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. Zbl0978.14033MR1768100
  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. Zbl0722.14013MR1106915
  19. —, Revêtement de la droite affine en caractéristique p &gt; 0 et conjecture d’Abhyankar. Invent. Math. 116 (1994), 425–462. Zbl0798.14013MR1253200
  20. —, Spécialisation des revêtements en caractéristique p &gt; 0 . Ann. Sci. École Norm. Sup. 32 (1999), no. 1, 87–126. MR1670532
  21. L. Schneps (ed.), The grothendieck theory of dessins d’enfants. London Math. Soc. Lecture Note Series, no. 200, Cambridge Univ. Press, 1994. Zbl0798.00001MR1305393
  22. H. Völklein, Groups as Galois groups. Cambridge Studies in Adv. Math., no. 53, Cambridge Univ. Press, 1996. Zbl0868.12003MR1405612
  23. S. Wewers, Formal deformations of curves with group scheme action. arXiv:math.AG/0212145, to appear in: Ann. Inst. Fourier. Zbl1079.14006MR2157165
  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. Zbl0995.14008MR1708609
  25. —, Reduction and lifting of special metacyclic covers. Ann. Sci. École Norm. Sup. 36 (2003), 113–138. Zbl1042.14005MR1987978
  26. —, Three point covers with bad reduction. J. Amer. Math. Soc. 16 (2003), no. 4, 991–1032. Zbl1062.14038MR1992833

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.