Semistable reduction and torsion subgroups of abelian varieties

Alice Silverberg; Yuri G. Zarhin

Annales de l'institut Fourier (1995)

  • Volume: 45, Issue: 2, page 403-420
  • ISSN: 0373-0956

Abstract

top
The main result of this paper implies that if an abelian variety over a field F has a maximal isotropic subgroup of n -torsion points all of which are defined over F , and n ≥ 5 , then the abelian variety has semistable reduction away from n . This result can be viewed as an extension of Raynaud’s theorem that if an abelian variety and all its n -torsion points are defined over a field F and n ≥ 3 , then the abelian variety has semistable reduction away from n . We also give information about the Néron models in the cases where n = 2 , 3 and 4.

How to cite

top

Silverberg, Alice, and Zarhin, Yuri G.. "Semistable reduction and torsion subgroups of abelian varieties." Annales de l'institut Fourier 45.2 (1995): 403-420. <http://eudml.org/doc/75123>.

@article{Silverberg1995,
abstract = {The main result of this paper implies that if an abelian variety over a field $F$ has a maximal isotropic subgroup of $n$-torsion points all of which are defined over $F$, and $n\ge 5$, then the abelian variety has semistable reduction away from $n$. This result can be viewed as an extension of Raynaud’s theorem that if an abelian variety and all its $n$-torsion points are defined over a field $F$ and $n\ge 3$, then the abelian variety has semistable reduction away from $n$. We also give information about the Néron models in the cases where $n=2,3$ and 4.},
author = {Silverberg, Alice, Zarhin, Yuri G.},
journal = {Annales de l'institut Fourier},
keywords = {abelian variety; -torsion points; semistable reduction; Néron models},
language = {eng},
number = {2},
pages = {403-420},
publisher = {Association des Annales de l'Institut Fourier},
title = {Semistable reduction and torsion subgroups of abelian varieties},
url = {http://eudml.org/doc/75123},
volume = {45},
year = {1995},
}

TY - JOUR
AU - Silverberg, Alice
AU - Zarhin, Yuri G.
TI - Semistable reduction and torsion subgroups of abelian varieties
JO - Annales de l'institut Fourier
PY - 1995
PB - Association des Annales de l'Institut Fourier
VL - 45
IS - 2
SP - 403
EP - 420
AB - The main result of this paper implies that if an abelian variety over a field $F$ has a maximal isotropic subgroup of $n$-torsion points all of which are defined over $F$, and $n\ge 5$, then the abelian variety has semistable reduction away from $n$. This result can be viewed as an extension of Raynaud’s theorem that if an abelian variety and all its $n$-torsion points are defined over a field $F$ and $n\ge 3$, then the abelian variety has semistable reduction away from $n$. We also give information about the Néron models in the cases where $n=2,3$ and 4.
LA - eng
KW - abelian variety; -torsion points; semistable reduction; Néron models
UR - http://eudml.org/doc/75123
ER -

References

top
  1. [1] B. BIRCH and W. KUYK, eds., Modular functions of one variable IV, Lecture Notes in Math. 476, Springer, New York, 1975, pp. 74-144. Zbl0315.14014
  2. [2] S. BOSCH, W. LÜTKEBOHMERT, M. RAYNAUD, Néron models, Springer, Berlin-Heidelberg-New York, 1990. Zbl0705.14001
  3. [3] B. EDIXHOVEN, Néron models and tame ramification, Comp. Math., 81 (1992), 291-306. Zbl0759.14033MR93a:14041
  4. [4] M. FLEXOR and J. OESTERLÉ, Sur les points de torsion des courbes elliptiques, Astérisque, Société Math. de France, 183 (1990), 25-36. Zbl0737.14004MR91g:11057
  5. [5] G. FREY, Some remarks concerning points of finite order on elliptic curves over global fields, Ark. Mat., 15 (1977), 1-19. Zbl0348.14018MR56 #15649
  6. [6] A. FRÖHLICH, Local fields, in Algebraic Number Theory, J. W. S. Cassels and A. Fröhlich, eds., Thompson Book Company, Washington, 1967, pp. 1-41. 
  7. [7] A. GROTHENDIECK, Modèles de Néron et monodromie, in Groupes de monodromie en géometrie algébrique, SGA7 I, A. Grothendieck, ed., Lecture Notes in Math. 288, Springer, Berlin-Heidelberg-New York, 1972, pp. 313-523. Zbl0248.14006MR50 #7134
  8. [8] H. LENSTRA and F. OORT, Abelian varieties having purely additive reduction, J. Pure and Applied Algebra, 36 (1985), 281-298. Zbl0557.14022MR86e:14020
  9. [9] D. LORENZINI, On the group of components of a Néron model, J. reine angew. Math., 445 (1993), 109-160. Zbl0781.14029MR94k:11065
  10. [10] H. MINKOWSKI, Gesammelte Abhandlungen, Bd. I, Leipzig, 1911, pp. 212-218 (Zur Theorie der positiven quadratischen Formen, J. reine angew. Math., 101 (1887), 196-202). JFM19.0189.01
  11. [11] D. MUMFORD, Abelian varieties, Second Edition, Tata Lecture Notes, Oxford University Press, London, 1974. 
  12. [12] D. MUMFORD, Tata Lectures on Theta II, Progress in Mathematics 43, Birkhäuser, Boston-Basel-Stuttgart, 1984. Zbl0549.14014
  13. [13] J-P. SERRE, Rigidité du foncteur de Jacobi d'echelon n ≥ 3, Appendix to A. Grothendieck, Techniques de construction en géométrie analytique, X. Construction de l'espace de Teichmüller, Séminaire Henri Cartan, 1960/1961, no. 17. 
  14. [14] J-P. SERRE and J. TATE, Good reduction of abelian varieties, Ann. of Math., 88 (1968), 492-517. Zbl0172.46101MR38 #4488
  15. [15] A. SILVERBERG and Yu. G. ZARHIN, Isogenies of abelian varieties, J. Pure and Applied Algebra, 90 (1993), 23-37. Zbl0832.14034MR94j:14040
  16. [16] J. H. SILVERMAN, The Néron fiber of abelian varieties with potential good reduction, Math. Ann., 264 (1983), 1-3. Zbl0497.14016MR85b:14059
  17. [17] A. WEIL, Variétés abéliennes et courbes algébriques, Hermann, Paris, 1948. Zbl0037.16202MR10,621d

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.