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Local Bounds for Torsion Points on Abelian Varieties

  • Pete L. Clark (a1) and Xavier Xarles (a2)

Abstract

We say that an abelian variety over a $p$ -adic field $K$ has anisotropic reduction $(\text{AR})$ 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 $\text{AR}$ 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 $\text{AR}$ 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.

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Local Bounds for Torsion Points on Abelian Varieties

  • Pete L. Clark (a1) and Xavier Xarles (a2)

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