论文标题
带有理性bitangents飞机四分之一的两种下降
Two-cover descent on plane quartics with rational bitangents
论文作者
论文摘要
我们在所有28个Bitangents合理的情况下实施了平面四分之一的两次下降，并表明在大量的测试用例集合中，它可以解决理性点的存在。我们还回顾了相关模量空间的经典描述，并使用它来生成示例。我们观察到，这种曲线对于局部障碍很少见，并且似乎只在减少的素数中才发生在实践中。特别是，在11个以11点降低意味着没有理性点。我们还收集了这些曲线的雅各布人两组等级的数值数据，这表明这些曲线通常具有非平凡的Tate-Shafarevich群体。 我们在所有28个Bitangents合理的情况下实施了平面四分之一的两次下降，并表明在大量的测试用例集合中，它可以解决理性点的存在。我们还回顾了相关模量空间的经典描述，并使用它来生成示例。我们观察到，这种曲线对于局部障碍很少见，并且在良好减少的素数中似乎只会发生。特别是，在11个以11点降低意味着没有理性点。我们还收集了这些曲线的雅各布人两组等级的数值数据，这提供了证据与亚伯利亚一般品种的行为不同，因为频繁存在局部琐碎的torsor。
We implement two-cover descent for plane quartics over Q with all 28 bitangents rational and show that on a significant collection of test cases, it resolves the existence of rational points. We also review a classical description of the relevant moduli space and use it to generate examples. We observe that local obstructions are quite rare for such curves, and only seem to occur in practice at primes of good reduction. In particular, having good reduction at 11 implies having no rational points. We also gather numerical data on two-Selmer ranks of Jacobians of these curves, which suggests that these often have non-trivial Tate-Shafarevich groups. We implement two-cover descent for plane quartics over Q with all 28 bitangents rational and show that on a significant collection of test cases, it resolves the existence of rational points. We also review a classical description of the relevant moduli space and use it to generate examples. We observe that local obstructions are quite rare for such curves and only seem to occur in practice at primes of good reduction. In particular, having good reduction at 11 implies having no rational points. We also gather numerical data on two-Selmer ranks of Jacobians of these curves, providing evidence these behave differently from those of general abelian varieties due to the frequent presence of an everywhere locally trivial torsor.
