Selmer Groups via 2-Isogenies on Quartic Curves Over Gaussian Integers — E8 Intelligence Research

FINDING: Selmer group computation for elliptic curves \(y^2 = x^3 + bx\) over \(Q(i)\) via graph-theoretic methods tied to 2-isogenies; average sizes of Selmer groups in isogeny families show deviation from naive heuristics, reflecting symmetry breaking in descent. MATH: - Curve: \(E_b: y^2 = x^3 + bx\), \(b ∈ Z[i]\), over \(Q(i)\). - Isogeny \(φ\) of degree 2: \(φ: E_b → E-4b\) (standard 2-isogeny for this form). - \(φ\)-Selmer group \(Sel(φ)(E_b/Q(i))\) — computed via weighted graph \(G_b\) with vertices = odd Gaussian primes \(p b\) (and possibly \(1+i\)). - Average size of Selmer group in families: for \(n+1\)-dimensional isogeny classes, heuristics (Poonen–Rains, Bhargava–Shankar) predict distribution; deviations occur when isogeny kernels intersect with \(n\)-torsion — e.g., \(#Sel(φ) ~ 2ʳ\) where \(r\) = rank of graph Laplacian over \(F_2\), not simply \ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com