Open problemChecked July 24, 2026

Birch and Swinnerton-Dyer Conjecture

A prediction connecting rational points on elliptic curves with their L-functions.

Number theory Arithmetic geometry Elliptic curves

Mathematical statement

For an elliptic curve $E/\mathbb{Q}$ , $\operatorname{rank}E(\mathbb{Q})=\operatorname{ord}_{s=1}L(E,s)$ .

Statement source: Therefore editorial statement

Statement terms: Source-specific

Editorial wording by Therefore. No statement reuse license is granted.

Statement artifacts, not proofs

These records expose exact Lean propositions and statement-only wrappers. Defining a proposition does not supply a proof of it. A placeholder-bearing target also contains no proof. Elaboration checks syntax and types; it does not certify that a formalization perfectly captures every nuance of the informal problem.

Pinned Lean formulation 1

MillenniumBirchSwinnertonDyer.clay_prize_birch_swinnerton_dyer

Canonical source

Complete statement target, proof intentionally absentLean 4

theorem clay_prize_birch_swinnerton_dyer :    ClayBirchSwinnertonDyer := by  sorry

Statement source: Lean Millennium Prize Problems
Lean version: v4.31.0
Placeholder: Present; no proof artifact
Source evidence: Pinned source index
Fidelity review: Community formulation

References

Clay Mathematics Institute problem pageofficial Birch and Swinnerton-Dyer overviewbackground