Infinite Families of CM Elliptic Curves Satisfy the 2-Part of BSD — E8 Intelligence Research

FINDING: The 2-part of BSD for quadratic twists of elliptic curves with complex multiplication is provably satisfied for infinite families of analytic rank 0 curves. | MATH: Let \(E/Q\) be an elliptic curve with complex multiplication (CM). For quadratic twists \(E⁽ᵈ⁾\), the analytic rank \(rₐₙ(E⁽ᵈ⁾) = 0\) is proven for infinitely many \(d\). The 2-part of the BSD conjecture (equality of order of \(Ш(E⁽ᵈ⁾)\) with the product of Tamagawa numbers, regulator, and \(L\)-value at 1) is then established for these twists. Key constants: \(L(E⁽ᵈ⁾,1) ≠ 0\) implies \(Ш(E⁽ᵈ⁾)[2^∞]\) is finite and its order matches the BSD formula. | CONNECTION: CM elliptic curves have endomorphism rings isomorphic to orders in imaginary quadratic fields, linking to lattice structures (root systems \(A_1\), \(D_4\) via the 2-torsion field). The quadratic twist families correspond to ray class fields, whose Galois groups are abelian and relate to base-60-like Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com