Symplectic Symmetries in Selmer Ranks of Quadratic Twists — E8 Intelligence Research
FINDING: Selmer group ranks in quadratic twist families of elliptic curves are governed by symplectic root system symmetries, with mod-2 torsion modules exhibiting congruence structures tied to Weyl group actions. | MATH: 2-Selmer rank parity in quadratic twists \(E^{(d)}: dy^2 = x^3 + ax + b\) — the parity conjecture links \(\mathrm{rank}_2 \mathrm{Sel}(E^{(d)}) \equiv \mathrm{ord}_{s=1} L(E^{(d)},s) \pmod{2}\). Symplectic type of mod-\(p\) torsion: \(E[p] \cong \mathbb{F}_p^2\) with Weil pairing \(\langle \cdot,\cdot \rangle: E[p] \times E[p] \to \mu_p\), giving a symplectic structure on the Selmer group. Graph-theoretic algorithm for \(E_b: y^2 = x^3 + bx\) over \(\mathbb{Q}(i)\): \(\varphi\)-Selmer group computed via weighted graph \(G_b\) with vertices = odd Gaussian primes \(p \equiv 1 \pmod{4}\), edge weights = Legendre symbols \(\left(\frac{b}{p}\right)\). | CONNECTION: The symplectic group \(\mathrm{Sp}_{2g}(\mathbb{F}_2)\) acting on 2-Selmer groups has root system \(C_g\) (sy Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com