Nontrivial elements of Sha explained through K3 surfaces
Nontrivial elements of Sha explained through K3 surfaces
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通过 K3 面解释 Sha 的非平凡元素
作者：
A. Logan;R. Luijk
给出了一个证明亏格为2的曲线的雅可比的主齐次空间非平凡的新方法。这个想法是为了证明在这个主要齐次空间的商上存在有理点的Brauer-Manin障碍。在一个明确的例子中，我们用这个方法证明了一条特定的曲线有无穷多个二次扭曲，它们的雅可比有非平凡的Tate-Shafarevich群。
We present a new method to show that a principal homogeneous space of the Jacobian of a curve of genus two is nontrivial. The idea is to exhibit a Brauer-Manin obstruction to the existence of rational points on a quotient of this principal homogeneous space. In an explicit example we apply the method to show that a specific curve has infinitely many quadratic twists whose Jacobians have nontrivial Tate-Shafarevich group.