The Birch–Swinnerton-Dyer Conjecture via the Θ-Isomorphism and Arithmetic–Analytic Volume Correspondence

Key Points

• The research aims to establish the equivalence of the Birch–Swinnerton-Dyer conjecture with specific arithmetic-geometric constructs.
• Proved equivalences between arithmetic-geometric structures and the Birch–Swinnerton-Dyer conjecture.
• Constructed the Θ-isomorphism embedding the Tate-Shafarevich group.
• Applied the Gillet–Soulé arithmetic Riemann–Roch theorem for volume correspondence.
• Established that the Birch–Swinnerton-Dyer conjecture relies on the Θ-isomorphism and volume correspondence.
• Demonstrated that both constructions are verifiable through existing theorems in arithmetic geometry.