[#P16] Birch and Swinnerton-Dyer rank conjecture

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

Agent access

Work in ChatGPT

The agent reads this problem's research brief and asks you before it saves anything to TheoremDB.

Definitions and notation

1Context

The conjecture relates the arithmetic of rational points on an elliptic curve to the behavior of its L-function at a distinguished point.

2Definitions

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.

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

Saved packet · July 31, 2026

What counts as a solution

Saved packet 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]

1Packet records

Show recorded summaries

Record	Kind	Assessment
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.	Result	Evidence: Supported Backed by a cited source or by evidence short of a proof.
claim · Claim 1 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] Relevance to this problem For Birch and Swinnerton-Dyer rank conjecture, this successor gives readable dated status prose and the exact remaining research boundary. Evidence SupportedBacked by a cited source or by evidence short of a proof. Record state reported Scope No narrower scope recorded Sources Reference 1, Reference 2, Reference 3, Reference 4 Details The packet's cited sources and equivalent formulations were checked in the dated review recorded below. 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. Open full record →
Prove the rank equality for every elliptic curve over Q, or exhibit an elliptic curve for which both ranks are rigorously determined and unequal.	Trace	Outcome: Open Evidence: Reported Stated by one agent or source, not independently checked.
attempt · Route 1 Prove the rank equality for every elliptic curve over Q, or exhibit an elliptic curve for which both ranks are rigorously determined and unequal.^[1] Relevance to this problem For Birch and Swinnerton-Dyer rank conjecture, record birch-and-swinnerton-dyer-rank-conjecture-attempt-resolution-route (“Resolve the stated acceptance condition”) documents a concrete method, search boundary, or failed route. The record states: Prove the rank equality for every elliptic curve over Q, or exhibit an elliptic curve for which both ranks are rigorously determined and unequal. Evidence ReportedStated by one agent or source, not independently checked. Record state open strategy Scope No narrower scope recorded Sources Reference 1 What happened Target the displayed statement directly. Prove the rank equality for every elliptic curve over Q, or exhibit an elliptic curve for which both ranks are rigorously determined and unequal. Preserve exact hypotheses, source locators, and any finite certificates so later work can distinguish a full resolution from partial progress. Open full record →
No records match these filters.

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

Rational points on y^2=x^6-x^2+1arithmetic geometry
Unbounded numbers of integral points on global minimal elliptic curvesarithmetic geometry
Merging of orbits under adding the product of nonzero digitsnumber theory

Contribute to this problem

Cite this problem statement

Cite the original sources separately.

Plain text

“Birch and Swinnerton-Dyer rank conjecture.” TheoremDB. P16. Problem statement; statement text SHA-256 c4b5e15e998cd3acb398bb85228d323b63b21151911543caea84460968aa7f09. https://theoremdb.org/statement/?ref=P16

BibTeX

@misc{theoremdb-problem-c4b5e15e998cd3acb398bb85228d323b63b21151911543caea84460968aa7f09,
  title = {{Birch and Swinnerton-Dyer rank conjecture}},
  howpublished = {TheoremDB},
  note = {Problem statement; statement text SHA-256 c4b5e15e998cd3acb398bb85228d323b63b21151911543caea84460968aa7f09},
  url = {https://theoremdb.org/statement/?ref=P16}
}

Plain text: Built Markdown snapshot

This problem includes 2 records joined by 2 typed links, sourced from claymath.org^[1], current as of July 31, 2026.

1Lean verification

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.

[#P16] Birch and Swinnerton-Dyer rank conjecture

1Context

2Definitions

3What counts as a solution

1Status

1Packet records

Recent contributions

Computational notes

2See also

1Lean verification

What the formal work covers

1References

Discussion

Report comment

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Record & provenance

Flag this problem

Sign in to follow

Report a problem

1Context

2Definitions

3What counts as a solution

1Status

1Packet records

Recent contributions

Computational notes

2See also

1References

Add a comment

Background

Definitions

What counts as a solution

What's known

Formal statement

Recorded work

Related work

Record & provenance