AI 中文总结
本文研究正则丢番图三元组诱导的椭圆曲线的算术，明确其挠子群结构，给出二次域上获3阶点的判定准则并验证特定族无此情况，还分析另一类D(-k²)三元组诱导曲线的挠与秩。
AI 中文摘要
我们研究由正则丢番图三元组诱导的椭圆曲线，重点关注其挠子群。证明由整数正则丢番图三元组诱导的椭圆曲线E，其挠子群必然满足E(Q)tors ≅ Z/2Z × Z/2Z。此外，我们给出此类椭圆曲线在二次域上获得3阶点的判定准则，并针对特定族{k-1, k+1, 4k}，运用该准则证明其不会出现该情况。最后，我们研究由D(-k²)三元组{1, 2k², 2k²+2k+1}诱导的椭圆曲线族的挠子群与一般秩。
英文摘要
We study elliptic curves induced by regular Diophantine triples, with emphasis on their torsion subgroups. We show that an elliptic curve $E$ induced by a regular Diophantine triple in integers necessarily has torsion subgroup $E(\mathbb{Q})_{\mathrm{tors}} \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$. Moreover, we develop a criterion for when such an elliptic curve acquires a point of order $3$ over a quadratic field. For a particular family $\{ k-1, k+1, 4k\}$, we use it to show that this does not happen. Finally, we study both the torsion and the generic rank of a family of elliptic curves induced by the $D(-k^2)$-triple $\{1, 2k^2, 2k^2+2k+1\}$.
Comments18 pages, code available at https://github.com/NikolaAdzaga/TorsionRegularTriples
