[#P16] Birch and Swinnerton-Dyer rank conjecture
Problem. For every elliptic curve \(E/\mathbb{Q}\), one has \(\operatorname{rank}E(\mathbb{Q})=\operatorname{ord}_{s=1}L(E,s)\).
1Context
The conjecture relates the arithmetic of rational points on an elliptic curve to the behavior of its L-function at a distinguished point.
2Problem setup
Definition 1 (E(Q). E(Q) is the abelian group of rational points on E; its rank is the number of independent infinite-order generators in its finitely generated decomposition.
Definition 2 (The analytic rank). The analytic rank is the order of the zero of L(E,s) at s = 1 after analytic continuation.
Remark 1. The conjecture relates the arithmetic of rational points on an elliptic curve to the behavior of its L-function at a distinguished point.
3What counts as a solution
- Prove the rank equality for every elliptic curve over Q, or exhibit an elliptic curve for which both ranks are rigorously determined and unequal.
1Status
Current status (Dated status and exact unresolved remainder). Unresolved in this packet after the dated source check. Strongest checked result: For elliptic curves over Q, Gross-Zagier and Kolyvagin, together with modularity, prove equality of algebraic and analytic rank and finiteness of the Tate-Shafarevich group when the analytic rank is 0 or 1. The rank equality for analytic rank at least 2 remains open in general. Exact unresolved remainder: Prove equality of algebraic and analytic rank for every elliptic curve over Q, or exhibit a curve for which both ranks are rigorously determined and unequal.[1][2][3][4]
1Records
Notes and companion material
Original intake status. The cited authoritative source listed this problem as unsolved when checked on 2026-07-31. This is an admin-curated seed record, not an independent exhaustive literature review.
- This seed states the rank equality. Consult the official description for the refined leading-coefficient formula and exact prize statement.
Computational notes
- Computations of algebraic and analytic ranks for individual curves test cases without resolving the universal claim.
2See also
- Unbounded numbers of integral points on global minimal elliptic curvesarithmetic geometry
- Rational points on y^2=x^6-x^2+1arithmetic geometry
- All nonnegative limits of normalized consecutive-prime gapsnumber theory
How to cite
TheoremDB contributors, “Birch and Swinnerton-Dyer rank conjecture,” TheoremDB research memory, snapshot of July 31, 2026. https://theoremdb.org/statements/birch-and-swinnerton-dyer-rank-conjectureThis page as plain text: birch-and-swinnerton-dyer-rank-conjecture.md
This problem includes 2 records joined by 2 typed links, sourced from claymath.org[1], current as of July 31, 2026.
1References
- Packet source. Clay Mathematics Institute, Birch and Swinnerton-Dyer Conjecture, official Millennium Prize Problem page, checked 2026-08-01. Official Problem Description by Andrew Wiles; listed under Unsolved Millennium Prize Problems. ↗website · primary source · checked 2026-07-31Source use: original summary.The cited authoritative source listed this problem as unsolved when checked on 2026-07-22. This is an admin-curated seed record, not an independent exhaustive literature review.Also cited at official problem description and unsolved classification.Also cited at Editorial research route recorded 2026-07-31.Source used to formulate or check the problem record.Source used to assess the problem's recorded status.Provides the authoritative formulation and current unsolved status.Source named by the research packet.
- Benedict H. Gross and Don B. Zagier, “Heegner points and derivatives ofL-series”. Inventiones Mathematicae 84(2) (1986), 225-320. DOI 10.1007/BF01388809. main theorem relating Heegner points and first derivatives of elliptic-curve L-series. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Supplies the analytic-rank-one input for the checked low-rank boundary.
- V A Kolyvagin, “FINITENESS OF $ E(\mathbf{Q})$ AND $ \textrm{Ø}(E,\mathbf{Q})$ FOR A SUBCLASS OF WEIL CURVES”. Mathematics of the USSR-Izvestiya 32(3) (1989), 523-541. DOI 10.1070/IM1989v032n03ABEH000779. finiteness and rank consequences for modular elliptic curves with analytic rank at most one. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Completes the checked rank-zero and rank-one consequence when combined with Gross-Zagier and modularity.
- Christophe Breuil, Brian Conrad, Fred Diamond, and Richard Taylor, “On the modularity of elliptic curves over 𝐐: Wild 3-adic exercises”. Journal of the American Mathematical Society 14(4) (2001), 843-939. DOI 10.1090/S0894-0347-01-00370-8. main modularity theorem for elliptic curves over Q. ↗journal article · primary source · checked 2026-08-01Source use: original summary.Extends the low-analytic-rank Gross-Zagier and Kolyvagin argument to every elliptic curve over Q.
An original CC0 restatement prepared by TheoremDB maintainers.