# zbMATH — the first resource for mathematics

Semistable reduction and torsion subgroups of abelian varieties. (English) Zbl 0818.14017
Summary: 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 $$n\geq 5$$, 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 $$n\geq 3$$, 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.

##### MSC:
 14K15 Arithmetic ground fields for abelian varieties 14G15 Finite ground fields in algebraic geometry 14G99 Arithmetic problems in algebraic geometry; Diophantine geometry 11G10 Abelian varieties of dimension $$> 1$$
Full Text:
##### References:
 [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. · Zbl 0315.14014 [2] S. BOSCH, W. LÜTKEBOHMERT, M. RAYNAUD, Néron models, Springer, Berlin-Heidelberg-New York, 1990. · Zbl 0705.14001 [3] B. EDIXHOVEN, Néron models and tame ramification, Comp. Math., 81 (1992), 291-306. · Zbl 0759.14033 [4] M. FLEXOR and J. OESTERLÉ, Sur LES points de torsion des courbes elliptiques, Astérisque, Société Math. de France, 183 (1990), 25-36. · Zbl 0737.14004 [5] G. FREY, Some remarks concerning points of finite order on elliptic curves over global fields, Ark. Mat., 15 (1977), 1-19. · Zbl 0348.14018 [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] 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. · Zbl 0248.14006 [8] H. LENSTRA and F. OORT, Abelian varieties having purely additive reduction, J. Pure and Applied Algebra, 36 (1985), 281-298. · Zbl 0557.14022 [9] D. LORENZINI, On the group of components of a Néron model, J. reine angew. Math., 445 (1993), 109-160. · Zbl 0781.14029 [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). · JFM 19.0189.01 [11] D. MUMFORD, Abelian varieties, Second Edition, Tata Lecture Notes, Oxford University Press, London, 1974. [12] D. MUMFORD, Tata lectures on theta II, Progress in Mathematics 43, Birkhäuser, Boston-Basel-Stuttgart, 1984. · Zbl 0549.14014 [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] J-P. SERRE and J. TATE, Good reduction of abelian varieties, Ann. of Math., 88 (1968), 492-517. · Zbl 0172.46101 [15] A. SILVERBERG and Yu. G. ZARHIN, Isogenies of abelian varieties, J. Pure and Applied Algebra, 90 (1993), 23-37. · Zbl 0832.14034 [16] J. H. SILVERMAN, The Néron fiber of abelian varieties with potential good reduction, Math. Ann., 264 (1983), 1-3. · Zbl 0497.14016 [17] A. WEIL, Variétés abéliennes et courbes algébriques, Hermann, Paris, 1948. · Zbl 0037.16202
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.