2-Parity Conjecture Proven for Elliptic Curves with Isomorphic 2-Torsion — E8 Intelligence Research

FINDING: The 2-parity conjecture is proven for elliptic curves with isomorphic 2-torsion, linking analytic root numbers to algebraic ranks mod 2. | MATH: Let \(E_1, E_2/\mathbb{Q}\) with \(E_1[2] \cong E_2[2]\) as Galois modules. The theorem states: \(\text{ord}_{s=1}L(E_1,s) \equiv \text{ord}_{s=1}L(E_2,s) \pmod{2} \iff \text{rank}(E_1) \equiv \text{rank}(E_2) \pmod{2}\). Equivalently, the root number \(W(E) = \pm 1\) satisfies \(W(E) = (-1)^{\text{rank}(E)}\) under this condition. The proof uses complete 2-descent (homogeneous spaces, Selmer groups \(\text{Sel}^{(2)}(E)\)) and the Cassels–Tate pairing, reducing the parity to a local root number product \(\prod_v W_v(E)\). | CONNECTION: The 2-torsion isomorphism forces a shared 2-adic lattice structure — the Galois action on \(E[2]\) corresponds to a crystallographic root system of type \(A_1 \times A_1\) (if all 2-torsion rational) or \(A_2\) (if a single rational 2-torsion point). The parity of rank mod 2 mirrors the parity of the d Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com