在具有j不变量1728的$\boldsymbol{\textit{Q}(i)}$上的精确秩为4和6的无穷多条椭圆曲线
Infinitely many elliptic curves over $\mathbb{Q}(i)$ of exact ranks 4 and 6 with $j$-invariant 1728
AI总结：
该研究在$\boldsymbol{\textit{Q}(i)}$上构造了$j$-不变量为1728、精确秩为4和6的无穷多条椭圆曲线，通过Kai定理和$[1+i]$-下降法确定秩，扩展了早期秩为2的研究策略。
AI中文摘要：
对于每个$r\in\{4,6\}$，我们在$\boldsymbol{\textit{Q}(i)}$上构造了一个显式的单参数椭圆曲线族，该族包含无穷多条两两非同构、真正定义在$\boldsymbol{\textit{Q}(i)}$上且$j$-不变量为1728、精确秩为$r$的曲线。我们构造了显式的$\boldsymbol{\textit{Q}(i)}$有理点以从下方估计秩。数域上线性模式素数值的Kai定理提供了具有受控局部行为的特化，使我们能通过$[1+i]$-下降法获得匹配的上界。该构造扩展了作者早期秩为2的论文的策略，用满足若干互补平方恒等式的二元线性形式系统替代对称高斯素数构型。在秩为6的情形中，附着于三个构造点和$(0,0)$的支撑向量张成了自对偶Reed-Muller码$\boldsymbol{\text{RM}}(1,3)$，该码也作为控制Selmer群的二次剩余拉普拉斯算子的核出现。
英文摘要：
For each $r\in\{4,6\}$, we construct an explicit one-parameter family of elliptic curves over $\mathbb{Q}(i)$ containing infinitely many pairwise nonisomorphic curves genuinely defined over $\mathbb{Q}(i)$ with $j$-invariant $1728$ and rank exactly $r$. We construct explicit $\mathbb{Q}(i)$-rational points to bound the ranks from below. Kai's theorem on prime values of linear patterns over number fields provides specializations with controlled local behavior, allowing us to obtain matching upper bounds via $[1+i]$-descent. The construction extends the strategy of the author's earlier rank-$2$ paper by replacing a symmetric Gaussian-prime configuration with systems of binary linear forms satisfying several complementary square identities. In the rank-$6$ case, the support vectors attached to the three constructed points and $(0,0)$ span the self-dual Reed-Muller code $\mathrm{RM}(1,3)$, which also occurs as the kernel of the quadratic-residue Laplacian governing the Selmer group.
