Recent Progress on the 2-Part of the Birch and Swinnerton-Dyer Conjecture for Quadratic Twists — E8 Intelligence Research

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, with recent progress on the 2-part for quadratic twists. MATH: - Elliptic curve: \( E: y^2 = x^3 + ax + b \), \( a,b ∈ Q \), discriminant \( Δ ≠ 0 \). - L-function: \( L(E,s) = ∏p Δ (1 - a_p p⁻ˢ + p¹⁻²ˢ)⁻¹ \), where \( a_p = p + 1 - #E(F_p) \). - BSD conjecture: \( ordₛ₌₁ L(E,s) = rank_Q E(Q) \). - Refined BSD: \( lims→ 1 L(E,s)/(s-1)^r = {Ω_E · Reg(E) · ∏_p c_p · #Ш(E)}{|E(Q)ₜₒᵣₛ|^2} \), where \( Ω_E \) is the real period, Reg is the regulator, \( c_p \) are Tamagawa numbers, Ш is the Tate-Shafarevich group. - Recent result (arXiv:1712.01271): For a large class of elliptic curves, explicit infinite families of quadr Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com