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 $\phi$-Selmer group of the elliptic curve $E_b:y^2=x^3+bx$ over $\mathbb{Q}(i)$, where $b\in \mathbb{Z}[i]$ and $\phi$ 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 $\phi$-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.

Updated title. 26 pages, 1 figure. Improved exposition and corrected typos throughout. Major additions include the mod 2 reduction of the graph $G_b$, an explicit calculation of the $\phi$-Selmer group when $b$ is a product of inert primes, and two new infinite families of elliptic curves over $\mathbb{Q}(i)$ with trivial rank