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SRM University AP（SRM大学AP校区）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究Q(√2)上的特定丢番图方程，利用二次整数环性质证明其有两组解，并据此推导相关椭圆曲线的莫德尔-魏尔群不含二阶有理点的结论。
AI 中文摘要
本文研究丢番图方程x₁³ - x₂²x₁ + 1 = 0，其中x₁∈Q(√2)，x₂∈Z[√2]。利用二次整数环Z[√2]的算术性质，结合范数论证、整除性性质及其单位群的显式描述，证明该方程恰有两组解：(x₁,x₂)=(-1,0)和(1,√2)。作为应用，考虑椭圆曲线族Cₘ:Y²=X³ - m²X + 1（m∈Z[√2]），推导出对所有m≠0,√2，莫德尔-魏尔群Cₘ(Q(√2))不含二阶有理点。
英文摘要
Let $\mathbb{K}=\mathbb{Q}(\sqrt{-d})\text{ or }\mathbb{Q}(\sqrt{d})$, where $d$ is a positive square-free integer, and denote by $\mathcal{O}_\mathbb{K}$ the ring of integers of $\mathbb{K}$. I investigate the solution of the equation $x_1^{3}-x_2^{2}x_1+1=0$ where $x_1\in \mathbb{K}$ and $x_2\in\mathcal{O}_\mathbb{K}$. The case $\mathbb{K}=\mathbb{Q}(\sqrt{d})$ for $d\equiv 2,3\pmod{4}$ faces infinite units that require a separate treatment. Using the arithmetic of the quadratic integer rings $\mathbb{Z}[\sqrt{d}]]$, together with norm arguments, divisibility properties, and the explicit structure of its unit group, I prove that the equation has exactly two solutions, namely $(x_1,x_2)=(-1,0)~\text{ and }~(1,\sqrt{2})$ for $\mathbb{K}=\mathbb{Q}(\sqrt{2})$ and one solution $(-1,0)$ for $\mathbb{K}=\mathbb{Q}(\sqrt{d})$ As an application, I consider the family of elliptic curves $C_m:Y^{2}=X^{3}-m^{2}X+1,~ m\in\mathcal{O}_\mathbb{K},$ and deduce that, for every $m\neq0,\sqrt{2}$ the Mordell--Weil group $C_m(\mathbb{K})$ contains no rational point of order two.
