论文标题
通过显式椭圆曲线研究希尔伯特的第十个问题
Studying Hilbert's 10th problem via explicit elliptic curves
论文作者
论文摘要
N.García-Fritz和H.Pasten表明，希尔伯特的第10个问题在数字字段的整数中无法解决，$ \ mathbb {q}（\ sqrt [3] {p}，\ sqrt {-q}）$ for Prime of Primes $ p $ P $ P $ $ P $。我们提高了它们的比例，并将其结果扩展到表单$ \ Mathbb {q}（\ sqrt [3] {p}，\ sqrt {dq}）$的数字字段的情况，其中$ d $属于一个明确的无正方形整体家族。我们通过使用多个椭圆形曲线来实现这一目标，并通过更直接的方法替换其Iwasawa理论论点。
N.García-Fritz and H.Pasten showed that Hilbert's 10th problem is unsolvable in the ring of integers of number fields of the form $\mathbb{Q}(\sqrt[3]{p},\sqrt{-q})$ for positive proportions of primes $p$ and $q$. We improve their proportions and extend their results to the case of number fields of the form $\mathbb{Q}(\sqrt[3]{p},\sqrt{Dq})$, where $D$ belongs to an explicit family of positive square-free integers. We achieve this by using multiple elliptic curves, and replace their Iwasawa theory arguments by a more direct method.
