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

Year: 2026
Type: preprint
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. Description 7: Version 7 extends the Cross-Realization Locking framework into a universal, zero-exception theory and provides a purely deductive algebraic proof for the rank 2 Birch and Swinnerton-Dyer leading term. Alongside the 60-digit Archimedean Green determinant locking (U_inf = 2 ± 3.5e-77) and memory-free cyclotomic character-twist CRT verification (U ≡ 2 mod 4301 across p in {11, 17, 23} on 226005.g3 with #III = 4), Version 7 introduces: (1) symmetric universal p-adic (Kudla-BDR exterior-power Selmer) and Archimedean (Kudla-Green current) structural engines; (2) the Main Theorem (Theorem 13), proving invariance of the global lattice index U = prod c_q * #III / t^2 across all rank 2 elliptic curves over Q with zero exceptions (covering CM curves, all torsion subgroups, all Tamagawa types, anomalous primes a_p ≡ 1 mod p, p | t boundaries, and supersingular primes a_p ≡ 0 mod p); and (3) a fully general, unconditional deductive proof derived purely from universal lattice primitives without reliance on numerical curve instances. Version 8 Consolidates the complete proof architecture and deductive hierarchy for the rank 2 Birch and Swinnerton-Dyer (BSD) leading-term identity. Alongside the 60-digit Archimedean Green determinant locking (U_\infty = 2 \pm 3.5 \times 10^{-77}) and memory-free cyclotomic character-twist CRT verification (U \equiv 2 \pmod{4301} across p \in \{11, 17, 23\} on 226005.g3 with \#\mathrm{III} = 4), Version 8 introduces five major advances: Unconditional p-Adic Closure ([T]): The p-adic leading-term identity \mathcal{L}_p(E,T)\vert{}_{T^2} = \mathrm{Reg}_p(E) \cdot U(E) is unconditionally closed as a theorem via Skinner–Urban (Iwasawa Main Conjecture), Bertolini–Darmon–Rotger (BDR diagonal cycles), and Bernardi–Perrin-Riou. Incoherent Siegel--Eisenstein Anchor: The Archimedean bridge (\mathrm{B3}' / S-b) is rigorously anchored to the central derivative of the genus-2 (\mathrm{Sp}_4) \emph{incoherent} Siegel--Eisenstein series \mathcal{E}^{(2)}_{\mathrm{incoh}}(\tau, s, \Phi), backed by the proved local place-by-place identities of Kudla (1997), Kudla–Rapoport–Yang (2004), Bruinier–Yang, Gross–Keating/Kitaoka, and Kudla–Rapoport (2000). Explicit K"unneth Projection ([T]): The exact Hodge--K"unneth decomposition of H^4(E^3) (b_4 = 15) isolates the 6-dimensional essential algebraic (2,2)-projector P_{\mathrm{ess}}^{(2,2)}, proving that the (1,1,2)-component H^1(E_1) \otimes H^1(E_2) \to \mathbb{Q} carries the exact 2\cdot\mathrm{rank} = 4 height-pairing structure. Cyclotomic 2\times 2 Galois Core & Trace-Zero Inversion ([T]): Identifies the cycle counterpart Z(t_i) as the Galois-conjugate orbit \{P_i^{\sigma_0^k}\} and proves that the character-sum Fourier inversion \frac{1}{d}\sum_m \Lambda_{\chi_m} = H_{ij}(0) = G_\infty reconstructs the exact 2\times 2 height Gram matrix even under trace-zero conditions (\Lambda_{\chi=\mathbf{1}} = 0, certified on 6355e1 and 389.a1 with saturation index m^2 = 9). Exact Isolation of Conjecture H ([A]): With all auxiliary algebraic, p-adic, K"unneth, and Fourier-inversion components proven as [T], the entire rank-2 BSD leading-term identity is rigorously isolated to a single global spectral primitive---Conjecture H (\mathrm{B3}').
DOI: 10.5281/zenodo.23405517
Language: English
Open Access status: green
OpenAlex ID: W7221898387
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