On the Brauer–Siegel ratio for abelian varieties over function fields
On the Brauer–Siegel ratio for abelian varieties
over function fields
复制标题
关于阿贝尔变种的布劳尔-西格尔比
DOI：
10.2140/ant.2019.13.1069
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发表时间：
2018
影响因子：
1.3
通讯作者：
Douglas Ulmer
中科院分区：
文献类型：
--
作者：
Douglas Ulmer
Hindry提出了一个类似的经典Brauer-Siegel定理的交换品种在全球领域。粗略地说，它说Mordell-Weil群的调节子和Tate-Shafarevich群的阶的乘积应该具有类似于指数微分高度的大小。Hindry-Pacheco和Griffon已经用分析技巧证明了函数域上某些椭圆曲线族的这一点。在这项工作中，我们的目标是证明类似的结果，更多的代数参数，即通过直接的方法，泰特Shafarevich组和监管机构。我们恢复Hindry-Pacheco和Griffon的结果，并将其扩展到新的家庭，包括家庭的高维阿贝尔品种。
Hindry has proposed an analogue of the classical Brauer-Siegel theorem for abelian varieties over global fields. Roughly speaking, it says that the product of the regulator of the Mordell-Weil group and the order of the Tate-Shafarevich group should have size similar to the exponential differential height. Hindry-Pacheco and Griffon have proved this for certain families of elliptic curves over function fields using analytic techniques. Our goal in this work is to prove similar results by more algebraic arguments, namely by a direct approach to the Tate-Shafarevich group and the regulator. We recover the results of Hindry-Pacheco and Griffon and extend them to new families, including families of higher-dimensional abelian varieties.