Exhibiting SHA[2] on hyperelliptic Jacobians
Exhibiting SHA[2] on hyperelliptic Jacobians
复制标题
在超椭圆雅可比行列式上展示 SHA[2]
DOI：
10.1016/j.jnt.2005.10.007
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发表时间：
2006
影响因子：
0.7
通讯作者：
E. V. Flynn
中科院分区：
文献类型：
--
作者：
Nils Bruin;E. V. Flynn
我们讨论的方法来计算在Shafarevich-Tate组的雅可比更高的属曲线，强调可视化的理论和实践。特别是对于超椭圆曲线，这通常使得能够计算雅可比矩阵的秩，即使当2-塞尔默界不严格限制秩时。这在以前只适用于少数特殊情况。对于属2的曲线，我们还证明了与度4德尔佩佐表面的连接，并显示如何在这些表面上的Brauer-Manin障碍可以用来计算成员的Shafarevich-Tate组的雅可比。我们得到一个显式参数化的无限族的亏格2曲线的雅可比矩阵有非平凡的成员的Shafarevich-Tate群。最后，我们证明了在一定的条件下，某些亏格为2的曲线的雅可比行列式的2阶上圈的可视化维数是4，而不是一般的界32。
We discuss approaches to computing in the Shafarevich–Tate group of Jacobians of higher genus curves, with an emphasis on the theory and practice of visualisation. Especially for hyperelliptic curves, this often enables the computation of ranks of Jacobians, even when the 2-Selmer bound does not bound the rank sharply. This was previously only possible for a few special cases. For curves of genus 2, we also demonstrate a connection with degree 4 del Pezzo surfaces, and show how the Brauer–Manin obstruction on these surfaces can be used to compute members of the Shafarevich–Tate group of Jacobians. We derive an explicit parametrised infinite family of genus 2 curves whose Jacobians have nontrivial members of the Shafarevich–Tate group. Finally, we prove that under certain conditions, the visualisation dimension for order 2 cocycles of Jacobians of certain genus 2 curves is 4 rather than the general bound of 32.