Exact Structural Decomposition of the Birch and Swinnerton-Dyer L-value for Rank 2 Elliptic Curves via Archimedean Green Functions

Abstract This paper presents a numerical and structural decomposition of the leading term of the L-function L''(E,1) for rank 2 elliptic curves over ℚ, in the context of the exact Birch and Swinnerton-Dyer (BSD) conjecture. 1. Empirical and Theoretical Decomposition Néron-Tate Regulator Decomposition: We empirically demonstrate that the global Néron-Tate regulator decomposes strictly into a purely continuous Archimedean component—the determinant of the Arakelov Green function matrix det(G_∞)—and an arithmetic correction factor arising from finite primes. BSD Identity: High-precision numerical experiments confirm the identity L''(1)/2 = Ω · det(G_∞) · (Reg / det(G_∞)) · ∏ c_p · (#Ш / t^2). Eisenstein-Kronecker Series: We provide a structural explanation for this phenomenon by analyzing Eisenstein-Kronecker series: we show that the vanishing of the holomorphic component at s=1 due to (m,n) → (-m,-n) parity symmetry rigorously forces the regulator into the non-holomorphic, real-valued Green function (log|σ|), thereby placing the s=1 BSD formula explicitly within the proven motivic framework of higher regulators (Goncharov, Samart). 2. Cross-Realization Identity and Version 5 Completion In Version 5, we complete the cross-realization identity between the Archimedean and p-adic leading terms. Recognizing the regulator not as a transcendental obstruction, but as an arithmetic descent pairing that cancels realization-dependent periods, we show that both (L''(1)/2) / (Ω · Reg) and L_p / Reg_p isolate the exact same global rational unit U = ∏ c_p · (#Ш / t^2). By introducing a memory-free cyclotomic character-twist inversion via Birch's lemma that bypasses the over 24–70 GiB modular symbol memory barrier at large conductors, we verify this identity across multiple primes for 389.a1 (U=1) and confirm the non-trivial Tate-Shafarevich case on 226005.g3 (#Ш=4, U=2) both Archimedeanly to 10^(-20) precision and p-adically at the good ordinary prime p=11 (U_11 ≡ 2 mod 11). This anchors the rational coefficient directly to the Iwasawa Main Conjecture and Cassels-Tate duality. Description v6: Version 6 formalizes the Cross-Realization Locking Theorem for rank 2 elliptic curves, uniting exact Archimedean Green function determinants (det(G_inf) to 60-digit precision) with a memory-free cyclotomic wild character-twist method. By bypassing traditional modular symbol memory barriers (RAM << 1 GiB), the work verifies the motivic unit U = prod c_p * (#III / t^2) = 2 on the rank 2 curve 226005.g3 (#III = 4) both Archimedeanly and p-adically via a Chinese Remainder Theorem locking (U ≡ 2 mod 4301 across p in {11, 17, 23}), alongside universal multi-curve basis saturation validation.