Spectral Determinants and the BSD Conjecture

This paper establishes a spectral-arithmetic identity at the central point s = 1 for elliptic curves E/ℚ, identifying a renormalized Fredholm-type spectral determinant D̃ₑ(1) with the full Birch–Swinnerton–Dyer (BSD) factor: D̃ₑ(1) = L^{(r)}(E, 1) / (r!·Ωₑ) · (#E(ℚ)tor)² / Regₑ · ∏{p∣N} cₚ(E) · |Sha(E/ℚ)|, where r is the analytic rank, Ωₑ the real period, Regₑ the Néron–Tate regulator, cₚ(E) the Tamagawa numbers, and Sha the Tate–Shafarevich group. The proof is organized through five structural “gates”: Gate L: Local spectral blocks match Tamagawa numbers, Gate K: Global normalization is fixed via anchor curves (e.g. 37a1), Gate H: Spectral statistics verified for numerical stability in twist families, Gates A2/A3: Operator–cohomology comparison with Bloch–Kato Selmer groups, Gates R3/R4: Global assembly identifies the determinant with the full BSD expression. A key outcome is the determination of the local p-adic unit uₚ purely from operator-theoretic structure—without invoking the Iwasawa Main Conjecture or Gross–Zagier–Kolyvagin results. The identity is verified unconditionally in families admitting either a suitable Heegner anchor or ordinary primes satisfying Iwasawa conditions. The work offers a novel spectral-analytic approach to deep arithmetic identities, bridging transfer operator theory and the arithmetic of elliptic curves. It complements the companion paper “Completed Zeta via Operator Blocks.”