Unifying Cassels-Tate Pairings on p-Selmer Groups via Albanese-Albanese and Cubic Norms — E8 Intelligence Research

FINDING: Explicit computation of Cassels-Tate pairings on p-Selmer groups (p=2,3) of elliptic curves, unifying Cassels' original pairing with the modern Albanese-Albanese definition, and extending to 3-isogeny Selmer groups via cubic norm equations. | MATH: Cassels-Tate pairing \( ·,· CT: Sel^p(E/K) × Sel^p(E/K) → Q/Z\); Selmer group \(Sel^φ(J,K) ⊆ H^1(G_K, J[φ])\) with local conditions; Albanese-Albanese definition (Poonen–Stoll) uses \(H^1(K, A) × H^1(K, Â) → Q/Z\); 3-isogeny descent: \(Sel^φ → Sel[3]\) improving rank bound; cubic norm equations \(NL/K(x) = c\) for cyclic cubic extensions. | CONNECTION: The Selmer group is a finite subgroup of a Galois cohomology group — its order is a power of p. For p=2, the pairing is alternating, giving a symplectic structure on a \(F_2\)-vector space — the dimension is even, and the group order is \(2²ᵍ\) where \(g\) is the "Selm Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com