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Local bounds for torsion points on abelian varieties. (English) Zbl 1204.11090
Summary: We say that an abelian variety over a $$p$$-adic field $$K$$ has anisotropic reduction (AR) if the special fiber of its Néron minimal model does not contain a nontrivial split torus. This includes all abelian varieties with potentially good reduction and, in particular, those with complex or quaternionic multiplication. We give a bound for the size of the $$K$$-rational torsion subgroup of a $$g$$-dimensional AR variety depending only on $$g$$ and the numerical invariants of $$K$$ (the absolute ramification index and the cardinality of the residue field). Applying these bounds to abelian varieties over a number field with everywhere locally anisotropic reduction, we get bounds which, as a function of g, are close to optimal. In particular, we determine the possible cardinalities of the torsion subgroup of an AR abelian surface over the rational numbers, up to a set of 11 values which are not known to occur. The largest such value is 72.

##### MSC:
 11G10 Abelian varieties of dimension $$> 1$$ 11G25 Varieties over finite and local fields 14K15 Arithmetic ground fields for abelian varieties
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