Local Bounds for Torsion Points on Abelian Varieties
Local Bounds for Torsion Points on Abelian Varieties
复制标题
阿贝尔簇上扭点的局部界限
DOI：
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发表时间：
2008
期刊：
影响因子：
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通讯作者：
X. Xarles
中科院分区：
文献类型：
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作者：
P. L. Clark;X. Xarles
我们说p-进域K上的阿贝尔簇具有各向异性约化（ ext{AR}）$如果其Néron极小模型的特殊纤维不包含非平凡分裂环面。这包括所有具有潜在良好约简的阿贝尔变种，特别是那些具有复数或四元数乘法的阿贝尔变种。本文给出了g维空间的K-有理扭子群的大小的一个界 ext{AR}$仅依赖于g$和K$的数值不变量（绝对分歧指数和剩余域的基数）。应用这些界限的阿贝尔品种在数域与无处不在的局部各向异性减少，我们得到的界限，作为一个函数的$g$，接近最优。特别地，我们确定了一个$的挠子群的可能的基数， ext{AR}$阿贝尔曲面上的有理数，高达一组11个值，这是不知道发生。最大值为72。
Abstract We say that an abelian variety over a $p$ -adic field $K$ has anisotropic reduction $( ext{AR})$ 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 $ ext{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 $ ext{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.