New Proof of the 2-Part of BSD for Quadratic Twists — E8 Intelligence Research

Abstract

FINDING: The Birch and Swinnerton-Dyer (BSD) conjecture links the algebraic rank of an elliptic curve (number of independent rational points) to the order of vanishing of its L-function at s=1; recent progress proves the 2-part of BSD for infinite families of quadratic twists with analytic rank 0. | MATH: Elliptic curve E/Q: y² = x³ + ax + b. L-function L(E,s) — analytic continuation via modularity. BSD: ord_{s=1} L(E,s) = rank(E(Q)). The 2-part: |Ш(E)[2]| = (∏ c_p · |E(Q)_tors|²) / |E(Q)|² · (leading coefficient of L at s=1) — refined via Tamagawa numbers c_p, torsion, and Tate-Shafarevich group Ш. Recent result (arXiv:1712.01271): for a large class of E, explicit infinite families of quadratic twists with analytic rank 0, and the 2-part of BSD holds for many — i.e., Ш[2] is computed exactly. | CONNECTION: Elliptic curves are 1-dimensional abelian varieties — their rational points form a finitely generated abelian group (Mordell-Weil). The L-function's critical point s=1 is the "cente Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com