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Local Bounds for Torsion Points on Abelian Varieties

Published online by Cambridge University Press:  20 November 2018

Pete L. Clark  and
Xavier Xarles
Pete L. Clark
Affiliation:
Department of Mathematics, University of Georgia, Athens, GA 30602, U.S.A. e-mail:, pete@math.uga.edu
Xavier Xarles
Affiliation:
Departament de Matemátiques, Universitsat Autónoma de Barcelona, Catahunya, Spain e-mail:, xarles@mat.uab.es
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Abstract

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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.

Keywords

11G1014K15
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 60 , Issue 3 , 01 June 2008 , pp. 532 - 555
Copyright
Copyright © Canadian Mathematical Society 2008

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