A graph-theoretic approach to computing Selmer groups of elliptic curves y² = x³ + bx over ℚ(i)

2024/10/30 by Kling, Anthony, Savoie, Ben
#05C90 (Secondary) #11G05 (Primary) 14H52 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2410.22714

Abstract

We develop a graph-theoretic algorithm to compute the φ-Selmer group of the elliptic curve E_b: y² = x³ + bx over ℚ(i), where b ∈ ℤ[i] and φ 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 φ-Selmer group of E_b when b is a product of inert primes, and we construct several infinite families of elliptic curves over ℚ(i) with trivial Mordell-Weil rank.

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