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: \( E: y^2 = x^3 + ax + b \), \( a,b \in \mathbb{Q} \), discriminant \( \Delta \neq 0 \). - L-function: \( L(E,s) = \prod_{p \nmid \Delta} (1 - a_p p^{-s} + p^{1-2s})^{-1} \), where \( a_p = p + 1 - \#E(\mathbb{F}_p) \). - BSD conjecture: \( \text{ord}_{s=1} L(E,s) = \text{rank}_\mathbb{Q} E(\mathbb{Q}) \). - Refined BSD: \( \lim_{s\to 1} \frac{L(E,s)}{(s-1)^r} = \frac{\Omega_E \cdot \text{Reg}(E) \cdot \prod_p c_p \cdot \#\text{Ш}(E)}{|E(\mathbb{Q})_{\text{tors}}|^2} \), where \( \Omega_E \) 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