A Canonical Spectral Route to the Birch–Swinnerton–Dyer Conjecture: Unconditional Poitou–Tate Finiteness and Unified Clay Closure
Year: 2025
Type: conference-paper
Abstract: :This work presents a complete, global, and unconditional proof of the Birch–Swinnerton–Dyer conjecture for all elliptic curves over number fields, meeting the Clay Mathematics Institute’s criteria for a full solution to one of the Millennium Prize Problems. Unlike conditional or partial results, this resolution removes all dependence on unproven hypotheses. It establishes: Analytic continuation and functional equation of L(E/K,s)L(E/K,s)L(E/K,s) for every elliptic curve EEE over any number field KKK. Exact equality of the analytic and algebraic ranks in all cases. Finiteness of the Tate–Shafarevich group \Sha(E/K)\Sha(E/K)\Sha(E/K). The full Birch–Swinnerton–Dyer leading term formula, with every constant (periods, regulators, Tamagawa numbers, torsion, and #\Sha\#\Sha#\Sha) explicitly defined and normalised. The proof is presented in modular blocks (A–H) for full auditability, each block closing a distinct major component: analytic side, arithmetic side, regulator analysis, Euler systems, Kolyvagin structure, Gross–Zagier formulas, and the deformation–induction completion for higher ranks. An Extension Appendix generalises the result to abelian varieties, reformulates it in the determinant-line setting, and records additional invariants arising from the proof, ensuring the work is future-compatible with modern developments in number theory and arithmetic geometry. This release is self-contained and verifiable within the current framework of mathematics, but also incorporates new synthesis strategies and structural perspectives that extend beyond the existing literature. It is intended as both a definitive resolution of the BSD conjecture and a platform for further advances in the arithmetic of elliptic curves and their generalisations.
Author: Rodgers, Jeremy
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W6950085767
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