Birch and Swinnerton-Dyer Conjecture

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cd millennium-birch-swinnerton-dyer &&
claude "make progress on this problem"

Statement

Let $E$ be an elliptic curve defined over $\mathbb{Q}$ , let $E(\mathbb{Q})$ be its finitely generated group of rational points, and let $L(E,s)$ be its Hasse–Weil $L$ -function.

The Birch and Swinnerton-Dyer conjecture asserts that

\operatorname{rank} E(\mathbb{Q}) = \operatorname{ord}_{s=1} L(E,s).

Thus the number of independent rational points predicted algebraically equals the order to which the analytic function vanishes at $s=1$ . The refined conjecture also identifies the first nonzero Taylor coefficient at $s=1$ using arithmetic invariants of $E$ , including its regulator, periods, local Tamagawa factors, torsion subgroup, and Tate–Shafarevich group.

Prove the conjecture in the precise form stated in the official problem description for all elliptic curves over $\mathbb{Q}$ .

Metadata

Subjects: number theory, arithmetic geometry

Provenance

statement: Project-authored summary based on Andrew Wiles's official Clay Mathematics Institute problem description; no Clay prose is reproduced verbatim.

Prize status

unsolved

Official page

https://www.claymath.org/millennium/birch-and-swinnerton-dyer-conjecture/

Partial progress

New unconditional cases, sharper reductions, independently reproducible computations, and well-explained failed approaches are valuable even when they do not resolve the full conjecture.

Research program

millennium_prize

Completion criteria

Establish the conjecture for every elliptic curve over the rational numbers, including the equality between algebraic rank and analytic rank. The refined leading-coefficient formula is part of the full conjectural picture; use the official description for its precise factors and normalizations.

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