论文标题
等源阿贝尔品种上的扭转点
Torsion points on isogenous abelian varieties
论文作者
论文摘要
为了研究Zannier的猜想，我们研究了主导基地的Abelian方案的不可切换亚变化，并包含一组Zariski致密的扭转点，这些扭转点位于成对的同源纤维上。如果在代数数字上定义了所有内容，并且Abelian方案具有最大变化，我们证明这种子变量的几何通用纤维是扭转圈的结合。我们继续证明了这种结果的完全或部分明确的版本，其中椭圆形曲线家族的纤维功能。最后，我们采用了Galateau-Martínez的最新结果，以在给定的同学类别中Manin-Mumford问题中最大扭转圈的数量获得统一的界限。为了获得证据，我们根据Lang，Serre，Tate和Hindry适应了该策略，该策略使用使用Galois自动形态，这些汽车对扭力作用作为家庭环境的同性恋。
Investigating a conjecture of Zannier, we study irreducible subvarieties of abelian schemes that dominate the base and contain a Zariski dense set of torsion points that lie on pairwise isogenous fibers. If everything is defined over the algebraic numbers and the abelian scheme has maximal variation, we prove that the geometric generic fiber of such a subvariety is a union of torsion cosets. We go on to prove fully or partially explicit versions of this result in fibered powers of the Legendre family of elliptic curves. Finally, we apply a recent result of Galateau-Martínez to obtain uniform bounds on the number of maximal torsion cosets in the Manin-Mumford problem across a given isogeny class. For the proofs, we adapt the strategy, due to Lang, Serre, Tate, and Hindry, of using Galois automorphisms that act on the torsion as homotheties to the family setting.
