Local Bounds for Torsion Points on Abelian Varieties

Abstract

AbstractWe say that an abelian variety over ap-adic fieldKhas anisotropic reduction (AR) if the special fiber of its Néronminimal 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 theK-rational torsion subgroup of ag-dimensional AR variety depending only ongand the numerical invariants ofK(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 ofg, 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.