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 \(\langle \cdot,\cdot \rangle_{CT}: Sel^p(E/K) \times Sel^p(E/K) \to \mathbb{Q}/\mathbb{Z}\); Selmer group \(Sel^\phi(J,K) \subseteq H^1(G_K, J[\phi])\) with local conditions; Albanese-Albanese definition (Poonen–Stoll) uses \(H^1(K, A) \times H^1(K, \hat{A}) \to \mathbb{Q}/\mathbb{Z}\); 3-isogeny descent: \(Sel^\phi \to Sel^{[3]}\) improving rank bound; cubic norm equations \(N_{L/K}(x) = c\) 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 \(\mathbb{F}_2\)-vector space — the dimension is even, and the group order is \(2^{2g}\) where \(g\) is the "Selm Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com