What question did this study set out to answer?

The aim is to provide a comprehensive proof of the Birch–Swinnerton-Dyer conjecture for rational elliptic curves.

August 7, 2026Open Access

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

Key Points

The aim is to provide a comprehensive proof of the Birch–Swinnerton-Dyer conjecture for rational elliptic curves.
Global proof methodology applied to all rational elliptic curves.
Utilization of generalized higher-order Gross–Zagier identities and related complex structures.
Formalization of proof in Lean for systematic verification.
Proved equality between algebraic and analytic rank for rational elliptic curves.
Established the exact leading-term BSD formula and finiteness of the Tate–Shafarevich group.

Abstract

Version 2 (August 5, 2026): This paper presents a full unconditional global proof of the Birch–Swinnerton-Dyer conjecture for all rational elliptic curves, superseding Version 1. The proof establishes two core results: (1) the equality between algebraic rank and analytic rank; (2) the exact leading-term BSD formula, including finiteness of the Tate–Shafarevich group. The argument relies on generalized higher-order Gross–Zagier identities, Nekovář Selmer complexes, multirank Euler systems, the GL(2) Iwasawa main conjecture, and Bloch–Kato cohomology. Attached to this release is complete Lean formalization source code for algorithmic verification. All reasoning uses only established theorems of arithmetic geometry with no unproven auxiliary hypotheses.

Ask AI

Helpful

Bookmark

View Full Paper