Proving the 2-Part of BSD for Infinite Families of Quadratic Twists — E8 Intelligence Research

Abstract

FINDING: The Birch and Swinnerton-Dyer (BSD) conjecture links the algebraic rank of an elliptic curve (number of rational points) to the order of zero of its L-function at s=1. Recent work proves the 2-part of BSD for infinite families of quadratic twists with analytic rank 0. MATH: - Elliptic curve E over ℚ: L(E,s) = Σ a_n n^{-s} (Dirichlet series). BSD predicts: ord_{s=1} L(E,s) = rank(E(ℚ)). - For quadratic twist E_d: L(E_d, s) has analytic rank 0 (non-vanishing at s=1). - Key result: For a large class of E, existence of infinite family of d such that L(E_d,1) ≠ 0, and the 2-primary part of Tate-Shafarevich group Ш(E_d)[2] is finite and satisfies BSD formula. - No explicit constants (0.382, 0.618, etc.) appear; the work uses p-adic methods, Iwasawa theory, and Heegner points. CONNECTION: - Weak geometric link: Elliptic curves are tori (genus 1 curves) with lattice structure ℂ/Λ. The L-function's functional equation relates to modular symmetry (Γ₀(N) congruence subgroups) Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com