Mazur猜想与Kummer曲面上及其超椭圆推广上的意外有理曲线
Mazur's Conjecture and An Unexpected Rational Curve on Kummer Surfaces and their Superelliptic Generalisations
AI总结：
本文证明了Mazur猜想的一个特殊情形，通过研究Kummer曲面上的奇异有理曲线并推广至更一般的超椭圆曲面，扩展了前人结果并给出了无限制条件下的证明。
摘要：
我们证明了关于有理点拓扑的Mazur猜想的一个特殊情形。设$E$为定义在$\mathbb{Q}$上且$j$-不变量为$1728$的椭圆曲线。对于一类由四次多项式对$E$进行二次扭曲的椭圆束，具有正秩纤维的射影直线上的有理点在实拓扑中是稠密的。这推广了Rohrlich和Kuwata-Wang针对二次和三次多项式获得的结果。为了进行证明，我们研究了与定义在$\mathbb{Q}$上的两条椭圆曲线的乘积相关联的Kummer曲面$K$上的一条高度奇异的有理曲线，该曲线此前曾出现在Mestre、Kuwata-Wang和Satgé的出版物中。我们通过寻找直接证明有理性的代数方程，以更简单的方式生成了这条曲线。我们发现，相同的方程在一类推广Kummer构造的更广泛曲面上也能产生有理曲线。这带来了除Mazur猜想之外的进一步应用，例如超椭圆曲线同时扭曲上有理点的存在性。最后，我们在对两条椭圆曲线的$j$-不变量没有任何限制的情况下，给出了针对Kummer曲面$K$的Mazur猜想的证明。
英文摘要：
We prove the following special case of Mazur's conjecture on the topology of rational points. Let $E$ be an elliptic curve over $\mathbb{Q}$ with $j$-invariant $1728$. For a class of elliptic pencils which are quadratic twists of $E$ by quartic polynomials, the rational points on the projective line with positive rank fibres are dense in the real topology. This extends results obtained by Rohrlich and Kuwata-Wang for quadratic and cubic polynomials. For the proof, we investigate a highly singular rational curve on the Kummer surface $K$ associated to a product of two elliptic curves over $\mathbb{Q}$, which previously appeared in publications by Mestre, Kuwata-Wang and Satgé. We produce this curve in a simpler manner by finding algebraic equations which give a direct proof of rationality. We find that the same equations give rise to rational curves on a class of more general surfaces extending the Kummer construction. This leads to further applications apart from Mazur's conjecture, for example the existence of rational points on simultaneous twists of superelliptic curves. Finally, we give a proof of Mazur's conjecture for the Kummer surface $K$ without any restrictions on the $j$-invariants of the two elliptic curves.
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