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Local Bounds for Torsion Points on Abelian Varieties
Published online by Cambridge University Press: 20 November 2018
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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- Research Article
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- Copyright © Canadian Mathematical Society 2008
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