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 \(\mathbb{Q}(i)\) 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: \(E_b: y^2 = x^3 + bx\), \(b \in \mathbb{Z}[i]\), over \(\mathbb{Q}(i)\). - Isogeny \(\varphi\) of degree 2: \(\varphi: E_b \to E_{-4b}\) (standard 2-isogeny for this form). - \(\varphi\)-Selmer group \(\mathrm{Sel}^{(\varphi)}(E_b/\mathbb{Q}(i))\) — computed via weighted graph \(G_b\) with vertices = odd Gaussian primes \(p \mid b\) (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., \(\#\mathrm{Sel}^{(\varphi)} \sim 2^{r}\) where \(r\) = rank of graph Laplacian over \(\mathbb{F}_2\), not simply \ Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com