Unifying Cassels-Tate Pairings on 2-Selmer Groups for Elliptic Curves — E8 Intelligence Research

FINDING: The Cassels-Tate pairing on 2-Selmer groups is identical to the classical Cassels-Tate pairing, unifying two approaches to measuring the failure of the Hasse principle for elliptic curves. | MATH: Let \(E/K\) be an elliptic curve, \(Sel_2(E/K)\) the 2-Selmer group. The classical Cassels-Tate pairing \(\langle \cdot,\cdot \rangle_{CT}: Sel_2(E/K) \times Sel_2(E/K) \to \mathbb{Q}/\mathbb{Z}\) is alternating and nondegenerate modulo the divisible part. Cassels' alternative pairing \(\langle \cdot,\cdot \rangle_{Cassels}\) (defined via local 2-descent maps and the Weil pairing \(e_2: E[2]\times E[2]\to \mu_2\)) is proven to satisfy \(\langle \cdot,\cdot \rangle_{Cassels} = \langle \cdot,\cdot \rangle_{CT}\). The kernel of the pairing is the maximal isotropic subgroup — precisely the image of the 2-Selmer group in the Tate-Shafarevich quotient, whose size relates to the 2-adic height regulator \(\det(\langle h_i,h_j \rangle_2)\) via the leading term of the \(L\)-function (Birch–Swi Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com