具有有理双切线的平面四次曲线上的双覆盖下降
Two-cover descent on plane quartics with rational bitangents
AI总结：
本文针对 Q 上 28 条双切线全为有理的平面四次曲线实现双覆盖下降，结合模空间构造算例，判定有理点存在性并分析局部障碍、双-Selmer 秩与 Tate–Shafarevich 群。
摘要：
我们对 Q 上所有 28 条双切线均为有理的平面四次曲线实现了双覆盖下降，并表明在大量测试案例集合中，该方法能够判定有理点的存在性。我们还回顾了相关模空间的经典描述，并利用它来生成例子。我们观察到，对于这类曲线，局部障碍相当罕见，而且在实践中似乎只出现在具有良好约化的素数处。特别地，在 11 处具有良好约化意味着不存在有理点。我们还收集了这些曲线的 Jacobian 的双-Selmer 秩的数值数据，这些数据表明它们通常具有非平凡的 Tate–Shafarevich 群。我们对 Q 上所有 28 条双切线均为有理的平面四次曲线实现了双覆盖下降，并表明在大量测试案例集合中，该方法能够判定有理点的存在性。我们还回顾了相关模空间的经典描述，并利用它来生成例子。我们观察到，对于这类曲线，局部障碍相当罕见，而且在实践中似乎只出现在具有良好约化的素数处。特别地，在 11 处具有良好约化意味着不存在有理点。我们还收集了这些曲线的 Jacobian 的双-Selmer 秩的数值数据，从而提供证据表明：由于处处局部平凡挠子的频繁存在，它们的行为不同于一般 Abel 簇。
英文摘要：
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.
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