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

Document Type

Journal Article

Role

Author

Published In

Journal of Number Theory

Volume

291

First Page

551

Last Page

591

Publication Date

9-25-2026

Abstract

We develop a graph-theoretic algorithm to compute the φ-Selmer group of the elliptic curve 𝐸𝑏 :𝑦2= 𝑥3+𝑏⁢𝑥 over ℚ⁡(𝑖), where 𝑏 ∈ℤ⁡[𝑖] and 𝜑 :𝐸𝑏 →𝐸−4⁢𝑏 is the degree 2 isogeny with kernel generated by (0,0). We associate to 𝐸𝑏 a weighted, directed graph 𝐺𝑏, 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. We then show that the local conditions defining the φ-Selmer group are encoded by a linear system over𝔽2 involving a modified Laplacian of𝐺𝑏. By applying our algorithm, we explicitly determine the φ-Selmer group of 𝐸𝑏 when every odd prime divisor of b is inert, and we construct several infinite families of elliptic curves overℚ⁡(𝑖) with Mordell–Weil rank 0.

Comments

Preprint version available via arXiv.

Keywords

Elliptic curves ; Selmer groups ; Descent ; Gaussian integers ; Ranks

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