Infinitely many elliptic curves over $\mathbb{Q}(i)$ with rank 2 and $j$-invariant 1728

Abstract

We prove that there exist infinitely many elliptic curves over $\mathbb{Q}(i)$ with $j$-invariant $1728$ and rank exactly $2$ which are not obtained by base change from $\mathbb{Q}$. The rank of each such curve is determined via 2-isogeny descent, and the existence of infinitely many such curves follows from Tao's constellation theorem for Gaussian primes.

8 pages. Introduced the notion of "elliptic curves genuinely defined over Q(i)" to reframe the main result. Removed secondary results (Theorems 1.3, 1.4 from v1) to shorten the paper. Streamlined several arguments, particularly in Section 2, and corrected minor typos