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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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