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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 \(> 1\)
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