Infinitely many elliptic curves over $\mathbb{Q}(i)$ of exact ranks 4 and 6 with $j$-invariant 1728

Abstract

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.