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

Abstract

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. We empirically demonstrate that the global Neron-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. High-precision numerical experiments confirm the identity L''(1)/2 = Ω · det(G_∞) · (Reg / det(G_∞)) · ∏c_p · (#UI / t²). Furthermore, 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). While exact deductive integration of q and the finiteness of the Tate-Shafarevich group remain open, this work successfully isolates and structurally explains the Archimedean Beilinson boundary.