A graph-theoretic approach to computing Selmer groups of elliptic curves $y^2 = x^3 + bx$ over $\mathbb{Q}(i)$

Abstract

We develop a graph-theoretic algorithm to compute the $\varphi$-Selmer group of the elliptic curve $E_b: y^2 = x^3 + bx$ over $\mathbb{Q}(i)$, where $b \in \mathbb{Z}[i]$ and $\varphi$ is a degree 2 isogeny of $E_b$. We associate to $E_b$ a weighted graph $G_b$, whose vertices are the odd Gaussian primes dividing $b$, and whose edge weights are determined by the quartic residue symbol between pairs of these primes. By applying our algorithm, we explicitly compute the $\varphi$-Selmer group of $E_b$ when $b$ is a product of inert primes, and we construct several infinite families of elliptic curves over $\mathbb{Q}(i)$ with trivial Mordell-Weil rank.