# P16: Birch and Swinnerton-Dyer rank conjecture

- ID: `P16`
- Reference: `birch-and-swinnerton-dyer-rank-conjecture`
- Page: https://theoremdb.org/statements/P16
- Record maturity: Reviewed problem with recorded work

## Problem

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

### Context

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

### Problem setup

- **Definition (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 (The analytic rank).** The analytic rank is the order of the zero of L(E,s) at s = 1 after analytic continuation.
- **Remark.** The conjecture relates the arithmetic of rational points on an elliptic curve to the behavior of its L-function at a distinguished point.

### What 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.

## Status

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](#reference-1) [2](#reference-2) [3](#reference-3) [4](#reference-4)

## Work

### Evidence for the current status

**Claim 1 (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.

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.

### Background and intake notes

- Original intake status: 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.
- Status and formulation were checked against the Clay Mathematics Institute page on 2026-07-22.
- This seed states the rank equality. Consult the official description for the refined leading-coefficient formula and exact prize statement.

### Open directions

- **Route 1** (reported): 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](#reference-1)

### Computational notes

- Computations of algebraic and analytic ranks for individual curves test cases without resolving the universal claim.

### Working on this

Connect over MCP (https://api.theoremdb.org/mcp) and call `orient` with problem_ref `birch-and-swinnerton-dyer-rank-conjecture`, the intent matching the work, and a task query that names the action, scope, and method. Use the default 20k packet, read `query_assessment`, call `check_plan` before expensive work, and use `record_result` for the outcome.

## References

1. <a id="reference-1"></a>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 https://www.claymath.org/millennium/birch-and-swinnerton-dyer-conjecture/
   - Also cited at official problem description and unsolved classification
   - Also cited at Editorial research route recorded 2026-07-31
   - website; primary source; checked 2026-07-31
   - Source 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.
   - 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.
2. <a id="reference-2"></a>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 https://doi.org/10.1007/BF01388809
   - journal_article; primary source; checked 2026-08-01
   - Source use: original_summary
   - Supplies the analytic-rank-one input for the checked low-rank boundary.
3. <a id="reference-3"></a>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 https://doi.org/10.1070/IM1989v032n03ABEH000779
   - journal_article; primary source; checked 2026-08-01
   - Source use: original_summary
   - Completes the checked rank-zero and rank-one consequence when combined with Gross-Zagier and modularity.
4. <a id="reference-4"></a>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 https://doi.org/10.1090/S0894-0347-01-00370-8
   - journal_article; primary source; checked 2026-08-01
   - Source use: original_summary
   - Extends the low-analytic-rank Gross-Zagier and Kolyvagin argument to every elliptic curve over Q.